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Matt Ridley

Matthew White Ridley, 5th Viscount Ridley, (born 7 February 1958), is a British science writer, journalist and businessman. He is known for his writings on science, the environment, and economics, and has been a regular contributor to The Times newspaper. Ridley was chairman of the UK bank Northern Rock from 2004 to 2007, during which period it experienced the first run on a British bank in 130 years. He resigned, and the bank was bailed out by the UK government; this led to its nationalisation.

Ridley is a libertarian, and a staunch supporter of Brexit. He inherited the viscountcy in February 2012 and was a Conservative hereditary peer from February 2013, with an elected seat in the House of Lords, until his retirement in December 2021.

Early life and education
Ridley's parents were Matthew White Ridley, 4th Viscount Ridley (1925–2012), and Lady Anne Katharine Gabrielle Lumley (1928–2006), the daughter of Roger Lumley, 11th Earl of Scarbrough. He is the nephew of the late Conservative Member of Parliament (MP) and minister Nicholas Ridley and the great grandson of Edwin Lutyens.

Ridley attended Eton College from 1970 to 1975, and then went on to Magdalen College, Oxford, to study zoology. Obtaining a BA degree with first class honours, Ridley continued with research on the mating system of the common pheasant (Phasianus colchicus) supervised by Chris Perrins for his Doctor of Philosophy degree in 1983.

Journalism
Ridley joined The Economist in 1984, first working as a science editor until 1987, then as Washington, D.C., correspondent from 1987 to 1989 and as American editor from 1990 to 1992. He was a columnist for The Daily Telegraph and The Sunday Telegraph and an editor of The Best American Science Writing 2002.

From 2010 to 2013, Ridley wrote the weekly "Mind and Matter" column for The Wall Street Journal, which "explores the science of human nature and its implications".

Since 2013, Ridley has written a weekly column for The Times on science, the environment, and economics.

Ridley wrote the majority of the main article of the August 2017 edition of BBC Focus magazine. The article explains his scepticism regarding resource depletion, challenging the widespread belief that resource depletion is an important issue. He cites various previous resource scares as his evidence.

Northern Rock, 1994–2007
In 1994, Ridley became a board member of the UK bank Northern Rock. His father had been a board member for 30 years, and chairman from 1987 to 1992. Ridley became chairman in 2004.

In September 2007, Northern Rock became the first British bank since 1878 to suffer a run on its finances, at the start of the financial crisis of 2007–2010. The bank applied to the Bank of England for emergency liquidity funding at the beginning of the financial crisis of 2007–08, but failed, and Northern Rock was nationalised. He resigned as chairman in October 2007. A parliamentary committee criticised Ridley for not recognising the risks of the bank's financial strategy and "harming the reputation of the British banking industry".

Business
From 1996 to 2003, Ridley served as founding chairman of the International Centre for Life, which opened in 2000 as a non-profit science centre in Newcastle upon Tyne; and is now its honorary life president. From July 2000 to June 2008, he was a non-executive director of PA Holdings Limited, with Victor Halberstadt.

Until 2010, he was a governor of the Ditchley Foundation, which organises conferences to further education and understanding of Britons and North Americans. He participated in a February 2000 Ditchley conference.

Patronage
The Banks Group and Blagdon estate developed and sponsored the construction of Northumberlandia, or the Lady of the North, a land sculpture in the shape of a reclining female figure, which was part-commissioned and sponsored by Ridley. Now run by a charity group called the Land Trust, it is the largest landform in the world depicting the human form, and, through private funding, cost £3m to build. Attracting over 100,000 people per year, the Northumberland art project, tourism and cultural landmark has won a global landscape architecture award, and has been named 'Miss World'.

The Royal Agricultural Society of England awarded the Bledisloe Gold Medal in 2015 to Ridley for the work done on his Blagdon estate, saying that it "wanted to highlight the extensive environmental improvement work that has been undertaken across the land".

Publications
Ridley has written a number of popular science books, listed below.

The Red Queen: Sex and the Evolution of Human Nature, 1993

 * In Lewis Carroll's Through the Looking-Glass, Alice meets the Red Queen who stays in the same place no matter how fast she runs. This book champions a Red Queen theory for the evolution of sexual reproduction: that it evolved so that the resultant genetic variation would thwart constantly mutating parasites.

The Origins of Virtue: Human Instincts and the Evolution of Cooperation, 1996

Genome: The Autobiography of a Species in 23 Chapters, 1999

 * This book examines one newly discovered gene from each of the 23 human chromosomes. It was shortlisted for the Samuel Johnson Prize in 2000.

Nature via Nurture: Genes, Experience, & What Makes Us Human, 2003 (also later released under the title The Agile Gene: How Nature Turns on Nurture in 2004)

 * This book discusses reasons why humans can be considered to be simultaneously free-willed and motivated by instinct and culture.

The Agile Gene: How Nature Turns on Nurture, 2004

Francis Crick: Discoverer of the Genetic Code, 2006

 * Ridley's biography of Francis Crick won the Davis Prize for the history of science from the US History of Science Society.

The Rational Optimist: How Prosperity Evolves, 2010

 * The Rational Optimist primarily focuses on the benefits of the innate human tendency to trade goods and services. Ridley argues that this trait is the source of human prosperity, and that as people increasingly specialize in their skill sets, we will have increased trade and even more prosperity. It was shortlisted for the 2011 BBC Samuel Johnson Prize.

The Evolution of Everything: How Ideas Emerge,    2015

 * In The Evolution of Everything, Ridley "makes the case for evolution, rather than design, as the force that has shaped much of culture, technology and society, and that even now is shaping our future." He argues that "Change in technology, language, mortality and society is incremental, inexorable, gradual and spontaneous...Much of the human world is the result of human action, but not of human design; it emerges from the interactions of millions, not from the plans of a few." The science writer Peter Forbes, writing in The Independent, describes the book as "Ridley's magnum opus, ... decades in the making." Forbes states that Ridley was inspired by the Roman poet Lucretius's long work on "atheistical atomism", De rerum natura, whose "arguments seem uncannily modern: like those of a Richard Dawkins 2000 years avant la lettre." Forbes found the chapter on technology to be "utterly convincing", the most satisfying in the book. But he finds the "sustained polemic on behalf of libertarian anti-State ideas not a million miles from those of the US Republican Tea Party." Forbes calls Ridley "a heretic on most counts", stating that the book has many excesses. All the same, he considers the book necessary reading.

How Innovation Works: And Why It Flourishes in Freedom, 2020

 * This book argues that innovation is a disorganized, bottom-up process that emerges through the aggregate work of many low-level individuals, rather than the work of solitary geniuses at the top. Moreover, innovation is poorly understood by economists, and it is often impeded by politicians. Ridley makes his case by examining historical examples, rather than appealing solely to abstract principles.

Viral: The Search for the Origin of COVID-19

 * Written jointly with Alina Chan, it was published in November 2021.

Ridley's first book was Warts and All: The Men Who Would Be Bush (1989), which chronicled the evolution of George H. W. Bush's public image during the 1988 United States presidential election. Ridley has since described his first book as "bad" and has expressed gratitude that few people know about it. He no longer promotes the book on his personal website.

In 2006, Ridley contributed a chapter to Richard Dawkins: How a Scientist Changed the Way We Think, a collection of essays in honour of his friend Richard Dawkins.

Ridley's 2010 TED conference talk, "When Ideas Have Sex", received over 2 million views. Ridley argues that exchange and specialisation are the features of human society that lead to the development of new ideas, and that human society is therefore a "collective brain".

Role of government regulation
In a 2006 edition of the online magazine Edge – the third culture, Ridley wrote a response to the question "What's your dangerous idea?" which was entitled "Government is the problem not the solution", in which he describes his attitude to government regulation: "In every age and at every time there have been people who say we need more regulation, more government. Sometimes, they say we need it to protect exchange from corruption, to set the standards and police the rules, in which case they have a point, though often they exaggerate it... The dangerous idea we all need to learn is that the more we limit the growth of government, the better off we will all be."

In 2007, the environmentalist George Monbiot wrote an article in The Guardian connecting Ridley's libertarian economic philosophy and the £27 billion failure of Northern Rock. On 1 June 2010 Monbiot followed up his previous article in the context of Matt Ridley's book The Rational Optimist, which had just been published. Monbiot took the view that Ridley had failed to learn from the collapse of Northern Rock.

Ridley has responded to Monbiot on his website, stating "George Monbiot's recent attack on me in the Guardian is misleading. I do not hate the state. In fact, my views are much more balanced than Monbiot's selective quotations imply." On 19 June 2010, Monbiot countered with another article on the Guardian website, further questioning Ridley's claims and his response. Ridley was then defended by Terence Kealey in a further article published on the Guardian website.

In November 2010, The Wall Street Journal published a lengthy exchange between Ridley and the Microsoft founder Bill Gates on topics discussed in Ridley's book The Rational Optimist. Gates said that "What Mr. Ridley fails to see is that worrying about the worst case—being pessimistic, to a degree—can actually help to drive a solution"; Ridley said "I am certainly not saying, 'Don't worry, be happy.' Rather, I'm saying, 'Don't despair, be ambitious.'"

Ridley summarised his own views on his political philosophy during the 2011 Hayek Lecture: "[T]hat the individual is not – and had not been for 120,000 years – able to support his lifestyle; that the key feature of trade is that it enables us to work for each other not just for ourselves; that there is nothing so anti-social (or impoverishing) as the pursuit of self sufficiency; and that authoritarian, top-down rule is not the source of order or progress."

In an email exchange, Ridley responded to the environmental activist Mark Lynas' repeated charges of a right-wing agenda with the following reply:

"On the topic of labels, you repeatedly call me a member of 'the right'. Again, on what grounds? I am not a reactionary in the sense of not wanting social change: I make this abundantly clear throughout my book. I am not a hierarchy lover in the sense of trusting the central authority of the state: quite the opposite. I am not a conservative who defends large monopolies, public or private: I celebrate the way competition causes creative destruction that benefits the consumer against the interest of entrenched producers. I do not preach what the rich want to hear—the rich want to hear the gospel of Monbiot, that technological change is bad, that the hoi polloi should stop clogging up airports, that expensive home-grown organic food is the way to go, that big business and big civil service should be in charge. So in what sense am I on the right? I am a social and economic liberal: I believe that economic liberty leads to greater opportunities for the poor to become less poor, which is why I am in favour of it. Market liberalism and social liberalism go hand in hand in my view."

Ridley argues that the capacity of humans for change and social progress is underestimated, and denies what he sees as overly pessimistic views of global climate change and Western birthrate decline.

Climate change
In 2014, the Wall Street Journal op-ed written by Ridley, "Whatever Happened to Global Warming?" suggesting that climate scientists' explanations were implausible, was challenged by Jeffrey Sachs of Columbia University's Earth Institute. Sachs termed "absurd" Ridley's characterization of a paper in Science magazine by the two scientists Xianyao Chen and Ka-Kit Tung. Sachs challenged Ridley's contentions, and claimed that the "paper's conclusions are the very opposite of Ridley's". Ridley replied that 'it is ludicrous, nasty and false to accuse me of lying or "totally misrepresenting the science..I have asked Mr. Sachs to withdraw the charges more than once now on Twitter. He has refused to do so ...."'

Friends of the Earth has connected Ridley's opposition to climate science to his ties to the coal industry. He is the owner of land in the north-east of England on which the Shotton Surface coal mine operates, and receives payments for the mine. In 2016 he was accused of lobbying for the coal industry, based on an email he had authored to the UK government's energy minister describing a Texas-based company which planned to sequester carbon into materials useful for industrial chemical manufacturing. The complaint was summarily dismissed by the Parliamentary Commissioner for Standards.

Shale gas and fracking
Ridley was one of the earliest commentators to spot the economic significance of shale gas. He is a proponent of fracking. However, he has been found to have breached the Parliamentary Code of Conduct by the House of Lords Commissioner for Standards for not orally disclosing in debates on the subject personal interests worth at least £50,000 in Weir Group, which has been described as "the world's largest provider of special equipment used in the process" of fracking.

Euroscepticism
Ridley is a Eurosceptic and advocated the withdrawal (Brexit) of the UK from the European Union during the 2016 United Kingdom European Union membership referendum. He appeared in Brexit: The Movie, arguing for Britain to return to the policy of free trade that distinguished it after 1845 until the 1930s.

Free-market anticapitalism
Ridley wrote a 2017 column making the case for free-market anticapitalism. He makes the case that it is misleading to refer to 'capitalism' and 'markets' as the same thing because "commerce, enterprise and markets are – to me – the very opposite of corporatism and even of 'capitalism', if by that word you mean capital-intensive organisations with monopolistic ambitions. Markets and innovation are the creative-destructive forces that undermine, challenge and reshape corporations and public bureaucracies on behalf of consumers. So big business is just as much the enemy as big government, and big business in hock to big government is sometimes the worst of all."

COVID-19
Ridley wrote in May 2020 that "research into the origins of the new coronavirus raises questions about how it became so infectious in human beings" and included as one possibility "perhaps laboratories". His 2021 book Viral: The Search for the Origin of COVID-19 written with Alina Chan, which received mixed reviews, ascribes the most likely proximate origin of the virus to the COVID-19 lab leak theory.

Honours, awards and titles
In 1996, he was a visiting professor at Cold Spring Harbor Laboratory in New York, and in 2006 was awarded an honorary DSc degree.

In 2003, he received an honorary DSc degree from Buckingham University and in 2007, an honorary DCL degree from Newcastle University.

In 2004, he was elected a Fellow of the Academy of Medical Sciences (FMedSci) for "major contributions to public engagement with the biological sciences".

In 2011, the Manhattan Institute awarded Ridley its $50,000 Hayek Prize for his book The Rational Optimist. In his acceptance speech, Ridley said: "As Hayek understood, it is human collaboration that is necessary for society to work... the key feature of trade is that it enables us to work for each other not just for ourselves; that attempts at self-sufficiency are the true form of selfishness as well as the quick road to poverty; and that authoritarian, top-down rule is not the source of order or progress." In 2011, Ridley gave the Angus Millar Lecture on "scientific heresy" at the Royal Society of Arts (RSA).

In 2012, on the death of his father, Ridley became the 5th Viscount Ridley and Baron Wensleydale. He is also the 9th Baronet Ridley. In 2013, he was elected as a hereditary peer to membership in the House of Lords, as a member of the Conservative Party.

In 2013, he was elected a member of the American Academy of Arts and Sciences, and won the Julian L. Simon award in March 2012. In 2014, he won the free enterprise award from the Institute of Economic Affairs.

Arms
As 5th Viscount Ridley, Matt Ridley bears arms blazoned as Gules on a Chevron Argent between three Falcons proper, as many Pellets.

Personal life
When his father died in 2012, Ridley succeeded him as the 5th Viscount Ridley, having taken over the running of the family estate of Blagdon Hall, near Stannington, Northumberland, some years before.

In 1989, Ridley married Anya Hurlbert, a Professor of Neuroscience at Newcastle University; they live in northern England and have a son and a daughter.

In 1980, his sister Rose married the British Conservative Party politician Owen Paterson, who held the posts of Secretary of State for Northern Ireland and Secretary of State for Environment, Food and Rural Affairs until July 2014. During this time Ridley was described as 'in many ways Paterson's personal think tank'.

In 2015, Ridley's team won the celebrity Christmas special of University Challenge representing Magdalen College, Oxford, the year after the team of his son, also Matthew, had won the student version representing Trinity College, Cambridge.

title: Matt Ridley


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Matrix addition

In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together.

For a vector, $$\vec{v}\!$$, adding two matrices would have the geometric effect of applying each matrix transformation separately onto $$\vec{v}\!$$, then adding the transformed vectors.

 * $$\mathbf{A}\vec{v} + \mathbf{B}\vec{v} = (\mathbf{A} + \mathbf{B})\vec{v}\!$$

However, there are other operations that could also be considered addition for matrices, such as the direct sum and the Kronecker sum.

Entrywise sum
Two matrices must have an equal number of rows and columns to be added. In which case, the sum of two matrices A and B will be a matrix which has the same number of rows and columns as A and B. The sum of A and B, denoted A + B, is computed by adding corresponding elements of A and B:

 * $$\begin{align}

\mathbf{A}+\mathbf{B} & = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{bmatrix} +

\begin{bmatrix} b_{11} & b_{12} & \cdots & b_{1n} \\ b_{21} & b_{22} & \cdots & b_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ b_{m1} & b_{m2} & \cdots & b_{mn} \\ \end{bmatrix} \\ & = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} & \cdots & a_{1n} + b_{1n} \\ a_{21} + b_{21} & a_{22} + b_{22} & \cdots & a_{2n} + b_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} + b_{m1} & a_{m2} + b_{m2} & \cdots & a_{mn} + b_{mn} \\ \end{bmatrix} \\

\end{align}\,\!$$ Or more concisely (assuming that A + B = C):
 * $$c_{ij}=a_{ij}+b_{ij}$$

For example:

\begin{bmatrix} 1 & 3 \\   1 & 0 \\    1 & 2  \end{bmatrix} + \begin{bmatrix} 0 & 0 \\   7 & 5 \\    2 & 1  \end{bmatrix} = \begin{bmatrix} 1+0 & 3+0 \\   1+7 & 0+5 \\    1+2 & 2+1  \end{bmatrix} = \begin{bmatrix} 1 & 3 \\   8 & 5 \\    3 & 3  \end{bmatrix} $$

Similarly, it is also possible to subtract one matrix from another, as long as they have the same dimensions. The difference of A and B, denoted A − B, is computed by subtracting elements of B from corresponding elements of A, and has the same dimensions as A and B. For example:

\begin{bmatrix} 1 & 3 \\ 1 & 0 \\    1 & 2 \end{bmatrix} - \begin{bmatrix} 0 & 0 \\ 7 & 5 \\ 2 & 1 \end{bmatrix} = \begin{bmatrix} 1-0 & 3-0 \\ 1-7 & 0-5 \\ 1-2 & 2-1 \end{bmatrix} = \begin{bmatrix} 1 & 3 \\ -6 & -5 \\ -1 & 1 \end{bmatrix} $$

Direct sum
Another operation, which is used less often, is the direct sum (denoted by ⊕). The Kronecker sum is also denoted ⊕; the context should make the usage clear. The direct sum of any pair of matrices A of size m × n and B of size p × q is a matrix of size (m + p) × (n + q) defined as:

\mathbf{A} \oplus \mathbf{B} = \begin{bmatrix} \mathbf{A} & \boldsymbol{0} \\ \boldsymbol{0} & \mathbf{B} \end{bmatrix} = \begin{bmatrix} a_{11} & \cdots & a_{1n} &     0 & \cdots &      0 \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ a_{m 1} & \cdots & a_{mn} &     0 & \cdots &      0 \\ 0 & \cdots &     0 & b_{11} & \cdots &  b_{1q} \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ 0 & \cdots &     0 & b_{p1} & \cdots &  b_{pq} \end{bmatrix} $$'''

For instance,

\begin{bmatrix} 1 & 3 & 2 \\   2 & 3 & 1  \end{bmatrix} \oplus \begin{bmatrix} 1 & 6 \\   0 & 1  \end{bmatrix} = \begin{bmatrix} 1 & 3 & 2 & 0 & 0 \\   2 & 3 & 1 & 0 & 0 \\    0 & 0 & 0 & 1 & 6 \\    0 & 0 & 0 & 0 & 1  \end{bmatrix} $$

The direct sum of matrices is a special type of block matrix. In particular, the direct sum of square matrices is a block diagonal matrix.

The adjacency matrix of the union of disjoint graphs (or multigraphs) is the direct sum of their adjacency matrices. Any element in the direct sum of two vector spaces of matrices can be represented as a direct sum of two matrices.

In general, the direct sum of n matrices is:

\bigoplus_{i=1}^{n} \mathbf{A}_{i} = \operatorname{diag}( \mathbf{A}_1, \mathbf{A}_2, \mathbf{A}_3, \ldots, \mathbf{A}_n) = \begin{bmatrix} \mathbf{A}_1 & \boldsymbol{0} & \cdots & \boldsymbol{0} \\ \boldsymbol{0} & \mathbf{A}_2 & \cdots & \boldsymbol{0} \\ \vdots & \vdots & \ddots & \vdots \\ \boldsymbol{0} & \boldsymbol{0} & \cdots & \mathbf{A}_n \\ \end{bmatrix}\,\!$$

where the zeros are actually blocks of zeros (i.e., zero matrices).

Kronecker sum
The Kronecker sum is different from the direct sum, but is also denoted by ⊕. It is defined using the Kronecker product ⊗ and normal matrix addition. If A is n-by-n, B is m-by-m and  $$\mathbf{I}_k$$ denotes the k-by-k identity matrix then the Kronecker sum is defined by:
 * $$ \mathbf{A} \oplus \mathbf{B} = \mathbf{A} \otimes \mathbf{I}_m + \mathbf{I}_n \otimes \mathbf{B}. $$

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Matrix addition


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USS Stark

USS Stark (FFG-31) was the 23rd ship of the Oliver Hazard Perry-class frigate of guided-missile frigates and was named after Admiral Harold Raynsford Stark (1880–1972). Ordered from Todd Pacific Shipyards in Seattle, Washington, on 23 January 1978, Stark was laid down on 24 August 1979, launched on 30 May 1980, and commissioned on 23 October 1982. In 1987, an Iraqi jet fired two missiles at Stark, killing 37 U.S. sailors on board. Decommissioned on 7 May 1999, Stark was scrapped in 2006.

Missile attack
USS Stark was deployed to the Middle East Force in 1984 and 1987. Captain Glenn R. Brindel was the commanding officer during the 1987 deployment. The ship was struck on 17 May 1987 by two Exocet anti-ship missiles during the Iran–Iraq War fired from an Iraqi aircraft officially identified as a Dassault Mirage F1 fighter, The Reagan administration attributed the blame to Iran for its alleged belligerence in the underlying conflict. The plane had taken off from Shaibah, Iraq at 20:00 and had flown south into the Persian Gulf. The pilot fired the first Exocet missile from a range of 22.5 nmi, and the second from 15.5 nmi, just about the time Stark issued a standard warning by radio. The frigate did not detect the missiles with radar; warning was given by the lookout only moments before the missiles struck. The first penetrated the port-side hull and failed to detonate, but left flaming rocket fuel in its path. The second entered at almost the same point, and, leaving a 3 by gash, exploded in crew quarters. The missiles killed 37 sailors and injured 21.

No weapons were fired in defense of Stark. The autonomous Phalanx CIWS remained in standby mode, Mark 36 SRBOC countermeasures were not armed until seconds before the missile hit. The attacking Exocet missiles and Mirage aircraft were in a blindspot of the STIR fire control director (Separate tracking and illumination Radar, part of the Mk 92 Guided Missile Fire Control System), and the Oto Melara Mk 75 76 mm/62 caliber naval gun, but in the clear for the MK 92 CAS (Combined Antenna System, primary search and tracking radar of the Mk 92 Guided Missile Fire Control System) and the Mk 13 Mod 4 single-arm launcher. The ship failed to maneuver to bring its Mk 75 to bear before the first missile hit.

On fire and listing, the frigate was brought under control by its crew during the night. The ship made its way to Bahrain where, after temporary repairs by the destroyer tender USS Acadia (AD-42) to make her seaworthy, she returned to her home port of Naval Station Mayport, under her own power. The ship was eventually repaired at Ingalls Shipbuilding in Mississippi for $142 million.

It is unknown whether Iraqi leaders authorized the attack. Initial claims by the Iraqi government that Stark was inside the Iran–Iraq War zone were shown to be false. The motives and orders of the pilot remain unanswered. American officials have claimed he was executed, but an ex-Iraqi Air Force commander later said that the pilot who attacked Stark was not punished, and remained alive. According to Jean-Louis Bernard, author of "Heroes of Bagdad" T1 (Editions JPO 2017), the pilot, Abdul Rhaman, not only was not punished, but would have received the medal of bravery at the end of 'a joint Iraqi-American commission of inquiry. His subsequent defection is not mentioned in this book. Jean-Louis Bernard also confirms the use of a Falcon 50 during this action. Citing lapses in training requirements and lax procedures, the U.S. Navy's board of inquiry relieved Captain Brindel of command and recommended him for court-martial, along with tactical action officer Lieutenant Basil E. Moncrief. Instead, Brindel and Moncrief received non-judicial punishment from Admiral Frank B. Kelso II and letters of reprimand. Brindel opted for early retirement while Moncrief resigned his commission after only eight years of service. The executive officer, Lieutenant Commander Raymond Gajan Jr., was detached for cause and received a letter of admonition.

1990s
Stark was part of the Standing Naval Forces Atlantic Fleet in 1990 before returning to the Middle East Force in 1991. Stark was attached to UNITAS in 1993 and took part in Operation Uphold Democracy and Operation Able Vigil in 1994. In 1995, Stark returned to the Middle East Force before serving with the Standing Naval Forces, Atlantic (STANAVFORLANT) in 1997 and in 1998.

Stark was decommissioned on 7 May 1999. A scrapping contract was awarded to Metro Machine Corp. of Philadelphia, Pennsylvania, on 7 October 2005. The ship was reported scrapped on 21 June 2006. Her stern plate was saved and donated to Naval Station Mayport.

title: USS Stark


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Hadamard product

In mathematics, the Hadamard product may refer to:
 * Hadamard product of two matrices, the matrix such that each entry is the product of the corresponding entries of the input matrices
 * Hadamard product of two power series, the power series whose coefficients are the product of the corresponding coefficients of the input series
 * a way of expressing an entire function of finite order
 * an infinite product expansion for the Riemann zeta function

title: Hadamard product


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Matrix multiplication

In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the first and the number of columns of the second matrix. The product of matrices $A$ and $B$ is denoted as $AB$.

Matrix multiplication was first described by the French mathematician Jacques Philippe Marie Binet in 1812, to represent the composition of linear maps that are represented by matrices. Matrix multiplication is thus a basic tool of linear algebra, and as such has numerous applications in many areas of mathematics, as well as in applied mathematics, statistics, physics, economics, and engineering. Computing matrix products is a central operation in all computational applications of linear algebra.

Notation
This article will use the following notational conventions: matrices are represented by capital letters in bold, e.g. $A$; vectors in lowercase bold, e.g. $a$; and entries of vectors and matrices are italic (they are numbers from a field), e.g. $A$ and $a$. Index notation is often the clearest way to express definitions, and is used as standard in the literature. The entry in row $i$, column $j$ of matrix $A$ is indicated by $(A)_{ij}$, $A_{ij}$ or $a_{ij}$. In contrast, a single subscript, e.g. $A_{1}, A_{2}$, is used to select a matrix (not a matrix entry) from a collection of matrices.

Matrix times matrix
If $A$ is an $m × n$ matrix and $B$ is an $n × p$ matrix, $$\mathbf{A}=\begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{pmatrix},\quad\mathbf{B}=\begin{pmatrix} b_{11} & b_{12} & \cdots & b_{1p} \\ b_{21} & b_{22} & \cdots & b_{2p} \\ \vdots & \vdots & \ddots & \vdots \\ b_{n1} & b_{n2} & \cdots & b_{np} \\ \end{pmatrix}$$ the matrix product $C = AB$ (denoted without multiplication signs or dots) is defined to be the $m × p$ matrix $$\mathbf{C} = \begin{pmatrix} c_{11} & c_{12} & \cdots & c_{1p} \\ c_{21} & c_{22} & \cdots & c_{2p} \\ \vdots & \vdots & \ddots & \vdots \\ c_{m1} & c_{m2} & \cdots & c_{mp} \\ \end{pmatrix}$$ such that $$ c_{ij} = a_{i1} b_{1j} + a_{i2} b_{2j} + \cdots + a_{in} b_{nj} = \sum_{k=1}^n a_{ik} b_{kj}, $$ for $i = 1, ..., m$ and $j = 1, ..., p$.

That is, the entry $c_{ij}$ of the product is obtained by multiplying term-by-term the entries of the $i$th row of $A$ and the $j$th column of $B$, and summing these $n$ products. In other words, $c_{ij}$ is the dot product of the $i$th row of $A$ and the $j$th column of $B$.

Therefore, $AB$ can also be written as $$\mathbf{C} = \begin{pmatrix} a_{11}b_{11} +\cdots + a_{1n}b_{n1} & a_{11}b_{12} +\cdots + a_{1n}b_{n2} & \cdots & a_{11}b_{1p} +\cdots + a_{1n}b_{np} \\ a_{21}b_{11} +\cdots + a_{2n}b_{n1} & a_{21}b_{12} +\cdots + a_{2n}b_{n2} & \cdots & a_{21}b_{1p} +\cdots + a_{2n}b_{np} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1}b_{11} +\cdots + a_{mn}b_{n1} & a_{m1}b_{12} +\cdots + a_{mn}b_{n2} & \cdots & a_{m1}b_{1p} +\cdots + a_{mn}b_{np} \\ \end{pmatrix} $$

Thus the product $AB$ is defined if and only if the number of columns in $A$ equals the number of rows in $B$, in this case $n$.

In most scenarios, the entries are numbers, but they may be any kind of mathematical objects for which an addition and a multiplication are defined, that are associative, and such that the addition is commutative, and the multiplication is distributive with respect to the addition. In particular, the entries may be matrices themselves (see block matrix).

Matrix times vector
A vector $$\mathbf x$$ of length $$n$$ can be viewed as a column vector, corresponding to an $$n\times1$$ matrix $$\mathbf X$$ whose entries are given by $$\mathbf X_{i1}=\mathbf x_i.$$ If $$\mathbf A$$ is an $$m\times n$$ matrix, the matrix-times-vector product denoted by $$\mathbf {Ax}$$ is then the vector $$\mathbf y$$ that, viewed as a column vector, is equal to the $$m\times1$$ matrix $$\mathbf{AX}.$$ In index notation, this amounts to:
 * $$y_i=\sum_{j=1}^n a_{ij}x_j.$$

One way of looking at this is that the changes from "plain" vector to column vector and back are assumed and left implicit.

Vector times matrix
Similarly, a vector $$\mathbf x$$ of length $$n$$ can be viewed as a row vector, corresponding to a $$1\times n$$ matrix. To make it clear that a row vector is meant, it is customary in this context to represent it as the transpose of a column vector; thus, one will see notations such as $$\mathbf{x}^\mathrm{T}\mathbf{A}.$$ The identity $$\mathbf{x}^\mathrm{T}\mathbf{A}=(\mathbf{A}^\mathrm{T}\mathbf{x})^\mathrm{T}$$ holds. In index notation, if $$\mathbf A$$ is an $$n\times p$$ matrix, $$\mathbf{x}^\mathrm{T}\mathbf{A}=\mathbf{y}^\mathrm{T}$$ amounts to: $$y_k=\sum_{j=1}^n x_j a_{jk}.$$

Vector times vector
The dot product $$\mathbf a\cdot\mathbf b$$ of two vectors $$\mathbf a$$ and $$\mathbf b$$ of equal length is equal to the single entry of the $$1\times 1$$ matrix resulting from multiplying these vectors as a row and a column vector, thus: $$\mathbf{a}^\mathrm{T}\mathbf{b}$$ (or $$\mathbf{b}^\mathrm{T}\mathbf{a},$$ which results in the same $$1\times 1$$ matrix).

Illustration

The figure to the right illustrates diagrammatically the product of two matrices $A$ and $B$, showing how each intersection in the product matrix corresponds to a row of $A$ and a column of $B$. $$ \overset{4\times 2 \text{ matrix}}{\begin{bmatrix} a_{11} & a_{12} \\ \cdot & \cdot \\ a_{31} & a_{32} \\ \cdot & \cdot \\ \end{bmatrix}} \overset{2\times 3\text{ matrix}}{\begin{bmatrix} \cdot & b_{12} & b_{13} \\ \cdot & b_{22} & b_{23} \\ \end{bmatrix}}

= \overset{4\times 3\text{ matrix}}{\begin{bmatrix} \cdot & c_{12} & \cdot \\ \cdot & \cdot & \cdot \\ \cdot & \cdot & c_{33} \\ \cdot & \cdot & \cdot \\ \end{bmatrix}} $$

The values at the intersections, marked with circles in figure to the right, are: $$\begin{align} c_{12} & = a_{11} b_{12} + a_{12} b_{22} \\ c_{33} & = a_{31} b_{13} + a_{32} b_{23}. \end{align}$$

Fundamental applications
Historically, matrix multiplication has been introduced for facilitating and clarifying computations in linear algebra. This strong relationship between matrix multiplication and linear algebra remains fundamental in all mathematics, as well as in physics, chemistry, engineering and computer science.

Linear maps
If a vector space has a finite basis, its vectors are each uniquely represented by a finite sequence of scalars, called a coordinate vector, whose elements are the coordinates of the vector on the basis. These coordinate vectors form another vector space, which is isomorphic to the original vector space. A coordinate vector is commonly organized as a column matrix (also called a column vector), which is a matrix with only one column. So, a column vector represents both a coordinate vector, and a vector of the original vector space.

A linear map $A$ from a vector space of dimension $n$ into a vector space of dimension $m$ maps a column vector
 * $$\mathbf x=\begin{pmatrix}x_1 \\ x_2 \\ \vdots \\ x_n\end{pmatrix}$$

onto the column vector
 * $$\mathbf y= A(\mathbf x)= \begin{pmatrix}a_{11}x_1+\cdots + a_{1n}x_n\\ a_{21}x_1+\cdots + a_{2n}x_n \\ \vdots \\ a_{m1}x_1+\cdots + a_{mn}x_n\end{pmatrix}.$$

The linear map $A$ is thus defined by the matrix
 * $$\mathbf{A}=\begin{pmatrix}

a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{pmatrix}, $$ and maps the column vector $$\mathbf x$$ to the matrix product
 * $$\mathbf y = \mathbf {Ax}.$$

If $B$ is another linear map from the preceding vector space of dimension $m$, into a vector space of dimension $p$, it is represented by a $p\times m$ matrix $$\mathbf B.$$ A straightforward computation shows that the matrix of the composite map $B\circ A$ is the matrix product $$\mathbf {BA}.$$ The general formula $(B\circ A)(\mathbf x) = B(A(\mathbf x))$) that defines the function composition is instanced here as a specific case of associativity of matrix product (see below):
 * $$(\mathbf{BA})\mathbf x = \mathbf{B}(\mathbf {Ax}) = \mathbf{BAx}.$$

Geometric rotations
Using a Cartesian coordinate system in a Euclidean plane, the rotation by an angle $$\alpha$$ around the origin is a linear map. More precisely, $$ \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos \alpha & - \sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix},$$ where the source point $$(x,y)$$ and its image $$(x',y')$$ are written as column vectors.

The composition of the rotation by $$\alpha$$ and that by $$\beta$$ then corresponds to the matrix product $$\begin{bmatrix} \cos \beta & - \sin \beta \\ \sin \beta & \cos \beta \end{bmatrix} \begin{bmatrix} \cos \alpha & - \sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix} = \begin{bmatrix} \cos \beta \cos \alpha - \sin \beta \sin \alpha & - \cos \beta \sin \alpha - \sin \beta \cos \alpha \\ \sin \beta \cos \alpha + \cos \beta \sin \alpha & - \sin \beta \sin \alpha + \cos \beta \cos \alpha \end{bmatrix} = \begin{bmatrix} \cos (\alpha+\beta) & - \sin(\alpha+\beta) \\ \sin(\alpha+\beta) & \cos(\alpha+\beta) \end{bmatrix},$$ where appropriate trigonometric identities are employed for the second equality. That is, the composition corresponds to the rotation by angle $$\alpha+\beta$$, as expected.

Resource allocation in economics
As an example, a fictitious factory uses 4 kinds of basic commodities, $$b_1, b_2, b_3, b_4$$ to produce 3 kinds of intermediate goods, $$m_1, m_2, m_3$$, which in turn are used to produce 3 kinds of final products, $$f_1, f_2, f_3$$. The matrices
 * $$\mathbf{A} = \begin{pmatrix} 1 & 0 & 1 \\  2 & 1 & 1 \\   0 & 1 & 1 \\   1 & 1 & 2 \\   \end{pmatrix} $$   and   $$\mathbf{B} = \begin{pmatrix} 1 & 2 & 1 \\   2 & 3 & 1 \\   4 & 2 & 2 \\   \end{pmatrix} $$

provide the amount of basic commodities needed for a given amount of intermediate goods, and the amount of intermediate goods needed for a given amount of final products, respectively. For example, to produce one unit of intermediate good $$m_1$$, one unit of basic commodity $$b_1$$, two units of $$b_2$$, no units of $$b_3$$, and one unit of $$b_4$$ are needed, corresponding to the first column of $$\mathbf{A}$$.

Using matrix multiplication, compute
 * $$\mathbf{AB} = \begin{pmatrix} 5 & 4 & 3 \\  8 & 9 & 5 \\\   6 & 5 & 3 \\   11 & 9 & 6 \\   \end{pmatrix} ;$$

this matrix directly provides the amounts of basic commodities needed for given amounts of final goods. For example, the bottom left entry of $$\mathbf{AB}$$ is computed as $$1 \cdot 1 + 1 \cdot 2 + 2 \cdot 4 = 11$$, reflecting that $$11$$ units of $$b_4$$ are needed to produce one unit of $$f_1$$. Indeed, one $$b_4$$ unit is needed for $$m_1$$, one for each of two $$m_2$$, and $$2$$ for each of the four $$m_3$$ units that go into the $$f_1$$ unit, see picture.

In order to produce e.g. 100 units of the final product $$f_1$$, 80 units of $$f_2$$, and 60 units of $$f_3$$, the necessary amounts of basic goods can be computed as
 * $$(\mathbf{AB}) \begin{pmatrix} 100 \\ 80 \\ 60 \\ \end{pmatrix} = \begin{pmatrix} 1000 \\ 1820 \\ 1180 \\ 2180 \end{pmatrix} ,$$

that is, $$1000$$ units of $$b_1$$, $$1820$$ units of $$b_2$$, $$1180$$ units of $$b_3$$, $$2180$$ units of $$b_4$$ are needed. Similarly, the product matrix $$\mathbf{AB}$$ can be used to compute the needed amounts of basic goods for other final-good amount data.

System of linear equations
The general form of a system of linear equations is
 * $$\begin{matrix}a_{11}x_1+\cdots + a_{1n}x_n=b_1,

\\ a_{21}x_1+\cdots + a_{2n}x_n =b_2, \\ \vdots \\ a_{m1}x_1+\cdots + a_{mn}x_n =b_m. \end{matrix}$$

Using same notation as above, such a system is equivalent with the single matrix equation
 * $$\mathbf{Ax}=\mathbf b.$$

Dot product, bilinear form and sesquilinear form
The dot product of two column vectors is the unique entry of the matrix product
 * $$\mathbf x^\mathsf T \mathbf y,$$

where $$\mathbf x^\mathsf T$$ is the row vector obtained by transposing $$\mathbf x$$. (As usual, a 1×1 matrix is identified with its unique entry.)

More generally, any bilinear form over a vector space of finite dimension may be expressed as a matrix product
 * $$\mathbf x^\mathsf T \mathbf {Ay},$$

and any sesquilinear form may be expressed as
 * $$\mathbf x^\dagger \mathbf {Ay},$$

where $$\mathbf x^\dagger$$ denotes the conjugate transpose of $$\mathbf x$$ (conjugate of the transpose, or equivalently transpose of the conjugate).

General properties
Matrix multiplication shares some properties with usual multiplication. However, matrix multiplication is not defined if the number of columns of the first factor differs from the number of rows of the second factor, and it is non-commutative, even when the product remains defined after changing the order of the factors.

Non-commutativity
An operation is commutative if, given two elements $A$ and $B$ such that the product $$\mathbf{A}\mathbf{B}$$ is defined, then $$\mathbf{B}\mathbf{A}$$ is also defined, and $$\mathbf{A}\mathbf{B}=\mathbf{B}\mathbf{A}.$$

If $A$ and $B$ are matrices of respective sizes $m\times n$ and $p\times q$, then $$\mathbf{A}\mathbf{B}$$ is defined if $n=p$, and $$\mathbf{B}\mathbf{A}$$ is defined if $m=q$. Therefore, if one of the products is defined, the other one need not be defined. If $m=q\neq n=p$, the two products are defined, but have different sizes; thus they cannot be equal. Only if $m=q= n=p$, that is, if $A$ and $B$ are square matrices of the same size, are both products defined and of the same size. Even in this case, one has in general
 * $$\mathbf{A}\mathbf{B} \neq \mathbf{B}\mathbf{A}.$$

For example
 * $$\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix},$$

but
 * $$\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}.$$

This example may be expanded for showing that, if $A$ is a $n\times n$ matrix with entries in a field $F$, then $$\mathbf{A}\mathbf{B} = \mathbf{B}\mathbf{A}$$ for every $n\times n$ matrix $B$ with entries in $F$, if and only if $$\mathbf{A}=c\,\mathbf{I}$$ where $c\in F$, and $I$ is the $n\times n$ identity matrix. If, instead of a field, the entries are supposed to belong to a ring, then one must add the condition that $c$ belongs to the center of the ring.

One special case where commutativity does occur is when $D$ and $E$ are two (square) diagonal matrices (of the same size); then $DE = ED$. Again, if the matrices are over a general ring rather than a field, the corresponding entries in each must also commute with each other for this to hold.

Distributivity
The matrix product is distributive with respect to matrix addition. That is, if $A, B, C, D$ are matrices of respective sizes $m × n$, $n × p$, $n × p$, and $p × q$, one has (left distributivity)
 * $$\mathbf{A}(\mathbf{B} + \mathbf{C}) = \mathbf{AB} + \mathbf{AC},$$

and (right distributivity)
 * $$(\mathbf{B} + \mathbf{C} )\mathbf{D} = \mathbf{BD} + \mathbf{CD}.$$

This results from the distributivity for coefficients by
 * $$\sum_k a_{ik}(b_{kj} + c_{kj}) = \sum_k a_{ik}b_{kj} + \sum_k a_{ik}c_{kj} $$
 * $$\sum_k (b_{ik} + c_{ik}) d_{kj} = \sum_k b_{ik}d_{kj} + \sum_k c_{ik}d_{kj}. $$

Product with a scalar
If $A$ is a matrix and $c$ a scalar, then the matrices $$c\mathbf{A}$$ and $$\mathbf{A}c$$ are obtained by left or right multiplying all entries of $A$ by $c$. If the scalars have the commutative property, then $$c\mathbf{A} = \mathbf{A}c.$$

If the product $$\mathbf{AB}$$ is defined (that is, the number of columns of $A$ equals the number of rows of $B$), then
 * $$ c(\mathbf{AB}) = (c \mathbf{A})\mathbf{B}$$ and $$ (\mathbf{A} \mathbf{B})c=\mathbf{A}(\mathbf{B}c).$$

If the scalars have the commutative property, then all four matrices are equal. More generally, all four are equal if $c$ belongs to the center of a ring containing the entries of the matrices, because in this case, $cX = Xc$ for all matrices $X$.

These properties result from the bilinearity of the product of scalars:
 * $$c \left(\sum_k a_{ik}b_{kj}\right) = \sum_k (c a_{ik} ) b_{kj} $$
 * $$\left(\sum_k a_{ik}b_{kj}\right) c = \sum_k a_{ik} ( b_{kj}c). $$

Transpose
If the scalars have the commutative property, the transpose of a product of matrices is the product, in the reverse order, of the transposes of the factors. That is
 * $$ (\mathbf{AB})^\mathsf{T} = \mathbf{B}^\mathsf{T}\mathbf{A}^\mathsf{T} $$

where T denotes the transpose, that is the interchange of rows and columns.

This identity does not hold for noncommutative entries, since the order between the entries of $A$ and $B$ is reversed, when one expands the definition of the matrix product.

Complex conjugate
If $A$ and $B$ have complex entries, then
 * $$ (\mathbf{AB})^* = \mathbf{A}^*\mathbf{B}^* $$

where $^{*}$ denotes the entry-wise complex conjugate of a matrix.

This results from applying to the definition of matrix product the fact that the conjugate of a sum is the sum of the conjugates of the summands and the conjugate of a product is the product of the conjugates of the factors.

Transposition acts on the indices of the entries, while conjugation acts independently on the entries themselves. It results that, if $A$ and $B$ have complex entries, one has

 * $$ (\mathbf{AB})^\dagger = \mathbf{B}^\dagger\mathbf{A}^\dagger ,$$

where $^{†}$ denotes the conjugate transpose (conjugate of the transpose, or equivalently transpose of the conjugate).

Associativity
Given three matrices $A, B$ and $C$, the products $(AB)C$ and $A(BC)$ are defined if and only if the number of columns of $A$ equals the number of rows of $B$, and the number of columns of $B$ equals the number of rows of $C$ (in particular, if one of the products is defined, then the other is also defined). In this case, one has the associative property
 * $$(\mathbf{AB})\mathbf{C}=\mathbf{A}(\mathbf{BC}).$$

As for any associative operation, this allows omitting parentheses, and writing the above products as $\mathbf{ABC}.$

This extends naturally to the product of any number of matrices provided that the dimensions match. That is, if $A_{1}, A_{2}, ..., A_{n}$ are matrices such that the number of columns of $A_{i}$ equals the number of rows of $A_{i + 1}$ for $i = 1, ..., n – 1$, then the product
 * $$ \prod_{i=1}^n \mathbf{A}_i = \mathbf{A}_1\mathbf{A}_2\cdots\mathbf{A}_n $$

is defined and does not depend on the order of the multiplications, if the order of the matrices is kept fixed.

These properties may be proved by straightforward but complicated summation manipulations. This result also follows from the fact that matrices represent linear maps. Therefore, the associative property of matrices is simply a specific case of the associative property of function composition.

Computational complexity depends on parenthezation
Although the result of a sequence of matrix products does not depend on the order of operation (provided that the order of the matrices is not changed), the computational complexity may depend dramatically on this order.

For example, if $A, B$ and $C$ are matrices of respective sizes $10×30, 30×5, 5×60$, computing $(AB)C$ needs $10×30×5 + 10×5×60 = 4,500$ multiplications, while computing $A(BC)$ needs $30×5×60 + 10×30×60 = 27,000$ multiplications.

Algorithms have been designed for choosing the best order of products, see Matrix chain multiplication. When the number $n$ of matrices increases, it has been shown that the choice of the best order has a complexity of $$O(n \log n).$$

Application to similarity
Any invertible matrix $$\mathbf{P}$$ defines a similarity transformation (on square matrices of the same size as $$\mathbf{P}$$)
 * $$S_\mathbf{P}(\mathbf{A}) = \mathbf{P}^{-1} \mathbf{A} \mathbf{P}.$$

Similarity transformations map product to products, that is
 * $$S_\mathbf{P}(\mathbf{AB}) = S_\mathbf{P}(\mathbf{A})S_\mathbf{P}(\mathbf{B}).$$

In fact, one has
 * $$\mathbf{P}^{-1} (\mathbf{AB}) \mathbf{P}

= \mathbf{P}^{-1} \mathbf{A}(\mathbf{P}\mathbf{P}^{-1})\mathbf{B} \mathbf{P} =(\mathbf{P}^{-1} \mathbf{A}\mathbf{P})(\mathbf{P}^{-1}\mathbf{B} \mathbf{P}).$$

Square matrices
Let us denote $$\mathcal M_n(R)$$ the set of $n×n$ square matrices with entries in a ring $R$, which, in practice, is often a field.

In $$\mathcal M_n(R)$$, the product is defined for every pair of matrices. This makes $$\mathcal M_n(R)$$ a ring, which has the identity matrix $I$ as identity element (the matrix whose diagonal entries are equal to 1 and all other entries are 0). This ring is also an associative $R$-algebra.

If $n > 1$, many matrices do not have a multiplicative inverse. For example, a matrix such that all entries of a row (or a column) are 0 does not have an inverse. If it exists, the inverse of a matrix $A$ is denoted $A^{−1}$, and, thus verifies
 * $$ \mathbf{A}\mathbf{A}^{-1} = \mathbf{A}^{-1}\mathbf{A} = \mathbf{I}. $$

A matrix that has an inverse is an invertible matrix. Otherwise, it is a singular matrix.

A product of matrices is invertible if and only if each factor is invertible. In this case, one has

 * $$(\mathbf{A}\mathbf{B})^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1}.$$

When $R$ is commutative, and, in particular, when it is a field, the determinant of a product is the product of the determinants. As determinants are scalars, and scalars commute, one has thus
 * $$ \det(\mathbf{AB}) = \det(\mathbf{BA}) =\det(\mathbf{A})\det(\mathbf{B}). $$

The other matrix invariants do not behave as well with products. Nevertheless, if $R$ is commutative, $AB$ and $BA$ have the same trace, the same characteristic polynomial, and the same eigenvalues with the same multiplicities. However, the eigenvectors are generally different if $AB ≠ BA$.

Powers of a matrix
One may raise a square matrix to any nonnegative integer power multiplying it by itself repeatedly in the same way as for ordinary numbers. That is,
 * $$\mathbf{A}^0 = \mathbf{I},$$
 * $$\mathbf{A}^1 = \mathbf{A},$$
 * $$\mathbf{A}^k = \underbrace{\mathbf{A}\mathbf{A}\cdots\mathbf{A}}_{k\text{ times}}.$$

Computing the $k$th power of a matrix needs $k – 1$ times the time of a single matrix multiplication, if it is done with the trivial algorithm (repeated multiplication). As this may be very time consuming, one generally prefers using exponentiation by squaring, which requires less than $2 log_{2} k$ matrix multiplications, and is therefore much more efficient.

An easy case for exponentiation is that of a diagonal matrix. Since the product of diagonal matrices amounts to simply multiplying corresponding diagonal elements together, the $k$th power of a diagonal matrix is obtained by raising the entries to the power $k$:

\begin{bmatrix} a_{11} & 0     & \cdots & 0 \\ 0     & a_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0    & 0       & \cdots & a_{nn} \end{bmatrix}^k = \begin{bmatrix} a_{11}^k & 0       & \cdots & 0 \\ 0       & a_{22}^k & \cdots & 0 \\ \vdots  & \vdots   & \ddots & \vdots \\ 0       & 0        & \cdots & a_{nn}^k \end{bmatrix}. $$

Abstract algebra
The definition of matrix product requires that the entries belong to a semiring, and does not require multiplication of elements of the semiring to be commutative. In many applications, the matrix elements belong to a field, although the tropical semiring is also a common choice for graph shortest path problems. Even in the case of matrices over fields, the product is not commutative in general, although it is associative and is distributive over matrix addition. The identity matrices (which are the square matrices whose entries are zero outside of the main diagonal and 1 on the main diagonal) are identity elements of the matrix product. It follows that the $n × n$ matrices over a ring form a ring, which is noncommutative except if $n = 1$ and the ground ring is commutative.

A square matrix may have a multiplicative inverse, called an inverse matrix. In the common case where the entries belong to a commutative ring $R$, a matrix has an inverse if and only if its determinant has a multiplicative inverse in $R$. The determinant of a product of square matrices is the product of the determinants of the factors. The $n × n$ matrices that have an inverse form a group under matrix multiplication, the subgroups of which are called matrix groups. Many classical groups (including all finite groups) are isomorphic to matrix groups; this is the starting point of the theory of group representations.

Matrices are the morphisms of a category, the category of matrices. The objects are the natural numbers that measure the size of matrices, and the composition of morphisms is matrix multiplication. The source of a morphism is the number of columns of the corresponding matrix, and the target is the number of rows.

Computational complexity

The matrix multiplication algorithm that results from the definition requires, in the worst case, $n^3$ multiplications and $(n-1)n^2$ additions of scalars to compute the product of two square $ω$ matrices. Its computational complexity is therefore $O(n^3)$, in a model of computation for which the scalar operations take constant time.

Rather surprisingly, this complexity is not optimal, as shown in 1969 by Volker Strassen, who provided an algorithm, now called Strassen's algorithm, with a complexity of $$O( n^{\log_{2}7}) \approx O(n^{2.8074}).$$ Strassen's algorithm can be parallelized to further improve the performance. , the best peer-reviewed matrix multiplication algorithm is by Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, and Renfei Zhou and has complexity $n×n$. It is not known whether matrix multiplication can be performed in $O(n^{2.371552})$ time. This would be optimal, since one must read the $f$ elements of a matrix in order to multiply it with another matrix.

Since matrix multiplication forms the basis for many algorithms, and many operations on matrices even have the same complexity as matrix multiplication (up to a multiplicative constant), the computational complexity of matrix multiplication appears throughout numerical linear algebra and theoretical computer science.

Generalizations
Other types of products of matrices include:
 * Block matrix operations
 * Cracovian product, defined as $n^{2 + o(1)}$
 * Frobenius inner product, the dot product of matrices considered as vectors, or, equivalently the sum of the entries of the Hadamard product
 * Hadamard product of two matrices of the same size, resulting in a matrix of the same size, which is the product entry-by-entry
 * Kronecker product or tensor product, the generalization to any size of the preceding
 * Khatri-Rao product and Face-splitting product
 * Outer product, also called dyadic product or tensor product of two column matrices, which is $$\mathbf{a}\mathbf{b}^\mathsf{T}$$
 * Scalar multiplication

title: Matrix multiplication


text:
Talk:Matrix multiplication

This article is just plain wrong
This article is based on the false assertion that a matrix is an "array of numbers". In its simplest case, a matrix IS a rectangular array of numbers, characterized by the number of its rows and columns, each of which is a positive integer. A matrix is always rectangular but it is NOT always an array of numbers. It can be an array of functions, sets, vectors, tensors, and most other mathematical objects, including matrices. Matrix multiplication is generally (as far as I know) defined as the direct generalization of the binary operation on numerical matrices, so it makes sense to explain it in those terms but to claim it is only about determining a numerical product array is just plain wrong.71.28.55.221 (talk) 18:55, 24 October 2016 (UTC)

 * The article says here that the entries do not have to be numbers, they could be functions or matrices as you point out. The definition of matrix multiplication as given in this article is correct. If you are talking about the lead,
 * "In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. Numbers such as the real or complex numbers can be multiplied according to elementary arithmetic. On the other hand, matrices are arrays of numbers, so there is no unique way to define "the" multiplication of matrices."
 * ...well... this is just a way to start from something the reader will most likely know before reading this article (multiplication of ordinary real numbers, possibly complex numbers). The rule for matrix multiplication does not change if the entries are functions or anything else. So what is your point? What does
 * "Matrix multiplication is generally (as far as I know) defined as the direct generalization of the binary operation on numerical matrices"
 * mean? 'M'∧Ŝc2ħεИτlk 19:47, 24 October 2016 (UTC)

I think that 71.* has a point and that the lead paragraph of the article is terrible. Actually it would make more sense (but not enough sense) if "matrices are arrays of numbers" was replaced by "matrices are not necessarily arrays of numbers". McKay (talk) 02:26, 25 October 2016 (UTC)

 * It is certainly true that matrices are not always numbers. I made a try at this; see what you think.   LouScheffer (talk) 12:22, 25 October 2016 (UTC)

 * Looks better, but if that was the only point, then its a bit overkill to claim the entire lead paragraph or even the entire article is "plain wrong"... 'M'∧Ŝc2ħεИτlk 17:34, 25 October 2016 (UTC)

 * But in this edit what is "each of which which makes sense in different contexts" supposed to mean? There are different definitions for multiplying matrices. It is unclear how they each "make sense in different contexts". One matrix multiplication may be useful for one purpose, another for another purpose, etc. so it would seem their usefulness, not "sense", is clearer. 'M'∧Ŝc2ħεИτlk 17:41, 25 October 2016 (UTC)

 * I would favor emphasizing the standard product first and only mentioning other types of matrix product later (say, in the second paragraph). That would correspond to how the body of the article is organized. Also, even though linear transformations are mentioned in the third paragraph, it is not mentioned that when two linear transformations are represented by matrices the composition of the transformations is represented by the product of the matrices.  This is surely the whole point of the standard matrix product. McKay (talk) 03:10, 26 October 2016 (UTC)

 * Makes sense to put the most common definition first, and explain why it is important. I tried this, feel free to change/comment.  LouScheffer (talk) 03:44, 26 October 2016 (UTC)

 * It's better. Now, what about the very first sentence: "In mathematics, matrix multiplication refers to any operation associated with the multiplication of matrix elements."? I don't think it makes sense.  (1) It doesn't say that there are two matrices involved. (2) It isn't true that any such operation is called "matrix multiplication"; only some of them are. Also, the parts of the third sentence of the first paragraph which refer to matrix size seem to just confuse the issue. I'm trying an entirely new paragraph; what do you think? McKay (talk) 01:04, 28 October 2016 (UTC)

 * Lead is looking better, good work from both of you. 'M'∧Ŝc2ħεИτlk 08:18, 29 October 2016 (UTC)

Scope?
If this article is to be about binary operations on two matrices to form a third, then scalar multiplication of matrices by a number should be excluded. Shall we transfer the section in this article to that article? The "see also" section of this article can link to the scalar multiplication article. It would prevent straining the lead between scalar multiplication and binary operations.

It could also make sense to have this article concentrate on the usual definition of the matrix product, with links to the other products (Hadamard product (matrices), Kronecker product etc.), and the Frobenius product split off to its own article (Frobenius product (matrices)) to be expanded later. 'M'∧Ŝc2ħεИτlk 08:18, 29 October 2016 (UTC)

 * I totally agree with restricting this article to the product of two matrices to form a third. Just because when one refers to "multiplication of a matrix/matrices", it is not clear which multiplication operation is being referred to does not mean that a single article should try to cover each of the myriad possibilities, as per WP:NAD.  —Quondum 15:32, 2 November 2016 (UTC)

 * Moved the scalar multiplication content over. It remains to decide if other matrix multiplications stay in this article or are removed altogether. 'M'∧Ŝc2ħεИτlk 09:55, 3 November 2016 (UTC)

 * IMO the "other" matrix multiplications should only be mentioned in "See also" section. The Frobenius inner product does not seem to have an article of its own yet, but the material should ideally not be lost.  The last two paragraphs of the lead do not belong there (strained, as you say, and conventions for an article should be explicated in the body, not the lead).  However, there seems to be a tendency to duplicate material in many articles (which I dislike if there is a clear place it "belongs"), so there may be resistance.  —Quondum 15:49, 3 November 2016 (UTC)

The article is now about the usual matrix product. 'M'∧Ŝc2ħεИτlk 10:28, 6 November 2016 (UTC)

 * Which is as it should be.   Nice to see articles being shaped into what they should be.  —Quondum 15:32, 6 November 2016 (UTC)

Needs early section on motivation
As someone mentioned in an earlier post, this article needs an early example (probably in a new section right after the lead, or in a new paragraph at the end of the lead) of why one would want to multiply matrices, in order to provide some context. Solving Ax=b occurs to me, but that requires using an inverse matrix, which would be too much for someone who is just beginning with matrices. Is there a simpler way to motivate matrix multiplication? Loraof (talk) 23:32, 22 December 2016 (UTC)

There is an applications section way down deep in the article, but it uses terminology that will be beyond the understanding of a beginner. It's good to have that section as is and where it is, but a more elementary example needs to be near the beginning. Loraof (talk) 23:39, 22 December 2016 (UTC)

 * Are you talking about Matrix multiplication? If so, adding a motivation section, when the lead already says linear transformations are one reason for the definition, will create duplication.

 * What could be done is a heuristic construction: start from the products of a row and column, then a square matrix and column, etc. at each stage explaining the purposes or examples (which the section already does). It would amount to putting Matrix multiplication before the general definition. 'M'∧Ŝc2ħεИτlk 10:36, 23 December 2016 (UTC)

 * No, Matrix multiplication just shows how to multiply matrices together. It does not say why we might want to do so. And while both that section and the lead mention the use in linear transformations, they don't say why we would want to do that. Loraof (talk) 14:47, 23 December 2016 (UTC)

 * Except the section does say why we want to do so. Please read what is under each example. 'M'∧Ŝc2ħεИτlk 10:02, 24 December 2016 (UTC)

Example
In the Dutch Wikipedia I found this example. I translated it, but it needs the hand of a native speaker.

Example
A company sells cement, chalk and plaster in bags weighing 25, 10, and 5 kg respectively. Four construction firms Arson, Build, Construct and Demolish, buy regularly these products from this company. The number of bags the clients buy in a specific year may be arranged in a 4×3-matrix A, with columns for the products and rows representing the clients:

 * $$A=\begin{bmatrix}14&9&3\\2&11&15\\0&12&17\\5&2&3\end{bmatrix}$$

We see for instance that $$A_{32}=12$$, indicating that client Construct has bought 12 bags of chalk that year.

A bag of cement costs GBP 12, a bag of chalk GBP 9 and a bag of plaster GBP 8. The 3×2-matrix B shows prices and weights of the three products:

 * $$B=\begin{bmatrix}12&25\\9&10\\8&5\end{bmatrix}$$

To find the total amount firm Arson has spent that year, we calculate:

 * $$14\times 12 + 9\times 9 + 3\times 8 =273$$,

in which we recognize the first row of the matrix A (Arson) and the first column of the matrix B (prices).

The total weight of the product bought by Arson is calculated in a similar manner:

 * $$14\times 25 + 9\times 10 + 3\times 5 =455$$,

in which we now recognize the first row of the matrix A (Arson) and the second column of the matrix B (weight).

We can make similar calculations for the other clients. Together they form the matrix AB as the matrix product of the matrices A and B:

 * $$AB=\begin{bmatrix}14&9&3\\2&11&15\\0&12&17\\5&2&3\end{bmatrix}

\begin{bmatrix}12&25\\9&10\\8&5\end{bmatrix}= \begin{bmatrix}273&455\\243&235\\244&205\\102&160\end{bmatrix} $$

Nijdam (talk) 11:55, 24 December 2016 (UTC)

 * Years ago there was probably an example like this. I prefer the generalized examples in Matrix multiplication, but if anyone wants a real-life example then it could be added. 'M'∧Ŝc2ħεИτlk 10:03, 24 December 2016 (UTC)

Indeed, it was me who proporsed it.Nijdam (talk) 11:55, 24 December 2016 (UTC)

I don't often edit wikipedia articles, so it's your call whether you want to insert this figure:
Use it if you think it makes the article better--Guy vandegrift (talk) 18:42, 8 February 2017 (UTC)
 * Note also that the muscle memory of your left-right hands reminds you that the left and right indices refer to first and second index respectively: rows and columns (from left-to-right). Advice offered by someone who might have a marginal case of dyslexia --Guy vandegrift (talk) 16:12, 10 February 2017 (UTC)

Pedantic question about the associative property
In the article, the product of row-vector with column-vector is used as an example of matrices product (also later, in the section The inner and outer products). Is that correct? Don't we get a problem with the associative property?

Say $A$ is a 1x3 matrix, $B$ is a 3x1 matrix and $C$ is a 5x5 matrix. In that case $(AB)C$ is defined, but $A(BC)$ isn't. Am I correct to say the associative property doesn't hold when the matrices are actually vectors? 79.179.82.237 (talk) 16:02, 23 August 2017 (UTC)

 * The confusion arises because strictly speaking $AB$ is a 1x1 matrix, not a scalar, so $(AB)C$ is strictly not defined. However, it is common to identify vectors of n components with 1xn matrices then ignore the difference between the 1x1 matrix whose single entry is the inner-product of the vectors, and the inner-product itself which is a scalar and not a matrix. McKay (talk) 05:40, 25 August 2017 (UTC)

 * Thank you for your answer, with which I agree completely.
 * Do you think we should add some remark/note, or am I just being fussy here? 79.179.82.237 (talk) 14:08, 26 August 2017 (UTC) — Preceding unsigned comment added by 84.229.130.80 (talk)

Incomprehensible Illustrations
Not only are there two illustrations on how to do matrix multiplication, the first one is a mess of arrows (and the image caption is no better), the second illustration is slightly better, but I had no idea that the two circles represent the dot products. If you do a google image search, you'll find far better illustrations, such as this one:

https://i1.wp.com/www.javatechblog.com/wp-content/uploads/2016/06/matrix-multiplication.jpg?resize=583%2C565

or that one:

https://www.mathsisfun.com/algebra/images/matrix-multiply-a.svg

The excessive use of symbols is not helping either (summations over indicides) is not helping either. Probably, the whole definition can be condensed to one picture and maybe a short reminder of the dot product. — Preceding unsigned comment added by 185.81.138.29 (talk) 17:23, 6 December 2017 (UTC)
 * I agree that the first figure is confusing. If one follows the arrows, one should believe that the sequence of operations is $((A_{i1}B_{1i}+A_{i2})B_{2i}+A_{i3})B_{3i}+\cdots.$ I'll remove this figure. D.Lazard (talk) 14:58, 20 February 2018 (UTC)

Math rating: Class = B ?
I have rewritten the whole article. For this, I have done some choices, generally explained in edit summaries. This rewriting is almost finished. However, a subsection of  is still needed, and I'll probably write it soon. In fact, it is now well known that the complexities of most problems of linear algebra are better expressed in terms of matrix-multiplication complexity, or in terms of the exponent $$\omega$$ of this complexity. It is thus useful to include that in WP, and presently, this article seems the best place.

IMHO, the rating of this article should be upgraded to "class = B". Being the author of the last major revision, I cannot upgrade it myself. So, your opinion is welcome, and if there is a consensus, for that, please, upgrade the rating. D.Lazard (talk) 18:14, 27 February 2018 (UTC)

 * Done. Thanks for your work on the article! Jakob.scholbach (talk) 20:39, 27 February 2018 (UTC)

Non-ring entries
"In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring."

What about other kinds of entries?

https://en.wikipedia.org/wiki/Logical_matrix

"Every logical matrix in U corresponds to a binary relation. These listed operations on U, and ordering, correspond to a calculus of relations, where the matrix multiplication represents composition of relations."

79.177.28.86 (talk) 06:16, 25 June 2018 (UTC)
 * Definitely should be expanded at least to semiring. Those are useful and standard e.g. for an algebraic view of graph shortest path problems (for which one wants the tropical semiring). But it's difficult to see how to go much more general than that and still have a useful notion of matrix multiplication. If it's just an array of things without a multiplication operation, it's not really a matrix. —David Eppstein (talk) 06:53, 25 June 2018 (UTC)

 * From article "Matrix"

 * "The entries need not be quadratic matrices, and thus need not be members of any ordinary ring; but their sizes must fulfil certain compatibility conditions."

 * (this is itself a wrong thing that a fact is mentioned in a more general article and not mentioned in a more speicial one)

 * This kind of structure to give entries for matrices to multiply is not limited to submatrices.

 * What's this kind of structure? It's a an analog of ring, with ring monoid replaced with a category.
 * 79.177.28.86 (talk) 01:27, 26 June 2018 (UTC)

German article and Manual of Style
Can I suggest people look at the German version (via Google translate) of this article and compare it to the English one?

I think the German version is a lot easier for someone unfamiliar with the topic to follow. I can see severel reasons.

Firstly, the German version is more inline with our MOS:MATH, which states:

 * "A general approach to writing an article is to start simple and then move towards more abstract and technical subjects later on in the article."

 * "The lead should, as much as possible, be accessible to a general reader, so specialized terminology and symbols should be avoided."

 * "Most mathematical ideas are capable of some form of generalization. If appropriate, such material can be put under a "Generalizations" section. As an example, multiplication of the rational numbers can be generalized to other fields."

This is not what has happend in this article, and I can see the history of why that is in some of the threads above. People coming to this talk page to insist some generalisation or other is mentioned in the first sentence. We end up with a first sentence that says

 * "In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring or even a semiring."

This is a really bad approach because it makes people think that they need to understand terms like field and semiring to understand matrix multiplication.

Relatedly, the German article talks about the sorts of things that are useful for a beginer to know, like "In order to be able to multiply two matrices, the number of columns of the first matrix must match the number of rows of the second matrix." It says this earlier and without the need to say "if $A$ is an $n × m$ matrix" etc.

Finally, the German article
 * makes much better use of images, including having one at the top of the article
 * devotes less space in the lead to compuational complexity
 * briefly mentions the history of matrix multiplication (something else recommended in MOS:MATH).

I would be happy to make some changes in line with the above, but I thought I should discuss it here first, given the previous conversations where people argue to include further generalisations in the lead.

Yaris678 (talk) 15:45, 13 December 2019 (UTC)

 * (following the note left at WT:WPM) Everything you say sounds very reasonable. The inclusion of jargon like "semiring" in the first sentence is particularly indefensible.  --JBL (talk) 22:24, 15 December 2019 (UTC)

 * Thanks JBL. I have now made a change addressing the above.  There are probably a few other pointers we cold take from the German article, like the use of further illustrations of the same format elsewhere in the article, but that will require further thought.  Yaris678 (talk) 14:43, 18 December 2019 (UTC)

 * I agree with removing semirings and such from the lead, especially the very first sentence. Another thing that should be improved is the fact that the lead does in no way summarize the article (as it should). Jakob.scholbach (talk) 08:36, 19 December 2019 (UTC)

Should the text in "Illustration" subsection of "Definition" actually be a caption of the illustration?
This section has been separated off from the section above
 * I didn't look at all the changes in detail, but the change to the example diagram was probably a bit much. What was there before may not be ideal, but I think trying to cram all of that into a caption will make things worse.  This appears to be an important enough example to have in the body of the article to help explain the process, and the diagram is just there as an extra visual aid to refer to.  –Deacon Vorbis (carbon • videos) 15:18, 18 December 2019 (UTC)
 * I hope you don't mind me making this a separate section. The issue here isn't really about the German article or our MOS.
 * I can see arguments both ways on this. It is basically an aesthetic question, rather than one of right and wrong.  But I think that making the text a caption makes it look neater and makes it clearer to the reader what the text is referring too.
 * I am happy to leave this aspect of the article as it is for now, but if any other editors have a view, I would appreciate reading it.
 * Yaris678 (talk) 15:38, 18 December 2019 (UTC)

Illustration of complexity
I'd like to thank for this change to the illustration on known computational complexity vs time. Beyond the reason he cites for the change (difficulty of updating the older image) the new one fixes a big problem with the older image: it showed the complexity as changing continuously and piecewise-linearly over time, but actually the change was discontinuous and piecewise-constant. —David Eppstein (talk) 21:36, 23 February 2020 (UTC)

Product matrix with vector
The article mentions the product of a matrix with a column vector, which is just a special case of the matrix product. It should also mention the product of a matrix with a vector, what is quite similar, but yet formally different. Otherwise there is a formal problem with the discussion of eigenvectors. Madyno (talk) 21:43, 3 September 2020 (UTC)
 * The product of a matrix with a vector cannot be defined if the coordinates of the vector are not defined. You are thinking of the application of a linear map on a vector, which is represented, when coordinates are defined, by the product of the matrix of the linear map by the column vector of the coordinates of the vector. D.Lazard (talk) 08:21, 4 September 2020 (UTC)

Right, with vector I meant an element of $$\R^n$$. Madyno (talk) 20:56, 5 September 2020 (UTC)

Pratical example
I like the example added by DeepKling. In fact, the article currently fails to give an idea of the purpose of matrix multiplication, and the example could fix this. I suggest to keep it and to add the tag, for now. In case it is in fact WP:OR, possible a similar example can be found in some introductory textbook, and then added with that source. Also, the presentation may be improved, possibly using an Svg picture (to be created). - Jochen Burghardt (talk) 21:28, 26 January 2022 (UTC)

 * We're not on a deadline, here. We can afford to wait for an example that actually has a source. MrOllie (talk) 21:30, 26 January 2022 (UTC)

 * Mmh, I think the problem is that I do not know of any textbook that provides an actual practical example, they just state how the calculation is done and I could not find an intuitive explanation on the internet as well. I always hate explanations where you think you would never come up with that idea yourself, creating an artificial complexity. However, I'm a little surprised that a simple example using the exact calculations that were introduced in the paragraph before and obviously producing the right result needs a source according to you. This is a little frustrating. I guess I either need to publish the example myself in a textbook or wait for someone else to bother writing such an example down ;-) . If I should create an svg, I can do so (I published several graphics on Wikipedia), but if any trivial example needs a source, I guess this would be pointless.--DeepKling (talk) 05:44, 28 January 2022 (UTC)

I found an example in a 1996 German textbook; here is an ad-hoc translation:

 A factory uses $$r_1, r_2, r_3, r_4$$ basic commodities (Rohstoffe) to produce $$z_1, z_2, z_3$$ intermediate goods, which in turn are used to produce $$e_1, e_2, e_3$$ final products. The matrices
 * $$A = \left( \begin{array}{ccc} 1 & 0 & 1 \\  2 & 1 & 1 \\   0 & 1 & 1 \\   1 & 1 & 2 \\   \end{array} \right)$$   and   $$B = \left( \begin{array}{ccc} 1 & 2 & 1 \\   2 & 3 & 1 \\   4 & 2 & 2 \\   \end{array} \right)$$

provide the amount of basic commodities needed for a given amount of intermediate goods, and the amount of intermediate goods needed for a given amount of final products, respectively. For example, to produce one unit of intermediate good $$z_2$$, two units of basic commodity $$r_1$$, one unit of $$r_2$$, and one unit of $$r_3$$ are needed, corresponding to the second row of $$A$$.

Using matrix multiplication, compute
 * $$A \cdot B = \left( \begin{array}{ccc} 5 & 4 & 3 \\  8 & 9 & 5 \\\   6 & 5 & 3 \\   11 & 9 & 6 \\   \end{array} \right)$$;

this matrixc directly provides the amounts of basic commodities needed for given amounts of final goods. For example, to produce 100 units of the final product $$e_1$$, 80 units of $$e_2$$, and 60 units of $$e_3$$, the necessary amounts of basic goods can be computed as
 * $$(AB) \left( \begin{array}{c} 100 \\ 80 \\ 60 \\ \end{array} \right) = \left( \begin{array}{c} 1000 \\ 1820 \\ 1180 \\ 2180 \\ \end{array} \right)$$,

that is, $$1000$$ units of $$r_1$$, $$1820$$ units of $$r_2$$, $$1180$$ units of $$r_3$$, $$2180$$ units of $$r_4$$ are needed. Similarly, the product matrix $$A \cdot B$$ can be used to compute the needed amounts of basic goods for other final-good amount data.

Maybe, this example could be used in the article. - Jochen Burghardt (talk) 08:46, 29 January 2022 (UTC)
 * This sort of examples is commonly encountered in operations research, specially in optimization of resource allocation through linear programming. One may certainly find similar examples in English textbooks on these subjects. D.Lazard (talk) 10:04, 29 January 2022 (UTC)

 * I gave it a try, and inserted the example (slighly adapted) in section Matrix_multiplication. I also added a brief example on rotation in Cartesian plane coordinates. Hope this is ok. - Jochen Burghardt (talk) 20:45, 2 February 2022 (UTC)

Ok, so not everything here is true…
See, the matrix multiplication ISN‘T binary. It is just a simple row of calculations and has been and is also used in analog computers, which are also NOT binary. That‘s it. This is the entire reason I wrote this. 91.181.173.247 (talk) 07:46, 21 May 2022 (UTC)

o(1)
In "It is not known whether matrix multiplication can be performed in $O(n^{2 + o(1)})$ time.", the formula seems wrong. $(2 + o(1))$ can be much larger than 2, which could result in $O(n^{3})$ or worse. The $o(1)$ should be removed like in the German article, or removed from the superscript: $O(n^{2} + o(1))$ Cymno (talk) 11:44, 26 July 2022 (UTC)
 * Incorrect. O-notation is only meaningful in the limit for large values of n. Every expression that can be bounded as $(2 + o(1))$ must be arbitrarily close to 2 for sufficiently large values of n. It is indeed unknown whether a bound of the form $O(n^{2 + o(1)})$ can be achieved. If you have edited the German article based on the same misunderstanding, perhaps you should undo your edits until you understand this material better. —David Eppstein (talk) 23:30, 26 July 2022 (UTC)
 * Isn't it possible to simplify the expression to avoid nested O(.) ? I have difficulties to understand what $$O(o(1))$$ is supposed to mean, let alone $$O(n^{2+o(1)})$$ - Jochen Burghardt (talk) 09:29, 27 July 2022 (UTC)
 * That's a good point, changed to $$n^{2+o(1)}$$ accordingly. $$O(n^{o(1)}) = n^{o(1)}$$, but that's not entirely obvious. Fawly (talk) 21:48, 27 July 2022 (UTC)
 * I agree that the outer O is redundant and better removed. —David Eppstein (talk) 23:04, 27 July 2022 (UTC)

AlphaTensor improved upon number of multiplications needed for matrix multiplications
Not sure if this is relevant, but a new AI by DeepMind managed to further improve upon previous lowest number of multiplications needed for matrix multiplication. Not sure if this actually leads to a decrease in time complexity though. 114.76.186.77 (talk) 05:23, 9 October 2022 (UTC)

 * Do you have a source? - Jochen Burghardt (talk) 11:30, 9 October 2022 (UTC)
 * This requires a reliable reference and a precise statement; at least, what is the number of multiplications for which dimensions of the matrices? D.Lazard (talk) 14:17, 9 October 2022 (UTC)
 * Yes, the papers have already been published
 * - The algorithm by DeepMind https://www.nature.com/articles/s41586-022-05172-4 (https://www.deepmind.com/blog/discovering-novel-algorithms-with-alphatensor)
 * - The algorithm by human beating DeepMind https://arxiv.org/abs/2210.04045 (https://www.newscientist.com/article/2341965-humans-beat-deepmind-ai-in-creating-algorithm-to-multiply-numbers/) Vip17 (talk) 13:42, 5 November 2022 (UTC)

Non-commutativity exceptions
I have reservations regarding the accuracy of the logical biconditional statement within the Non-commutativity sub-section, specifically the assertion that: if $A$ is a $n\times n$ matrix with entries in a field $F$, then $$\mathbf{A}\mathbf{B} = \mathbf{B}\mathbf{A}$$ for every $n\times n$ matrix $B$ with entries in $F$, if and only if $$\mathbf{A}=c\,\mathbf{I}$$ where $c\in F$, and $I$ is the $n\times n$ identity matrix."

I believe it should be expressed as a unidirectional implication, where the statement holds when $$\mathbf{A}=c\,\mathbf{I} \vee \mathbf{B}=c\,\mathbf{I}$$. Indeed, the biimplication is in contrast with an assertion made a few lines below, which states that: "One special case where commutativity does occur is when $D$ and $E$ are two (square) diagonal matrices (of the same size); then $DE = ED$".

Furthermore, matrix multiplication is commutative in other scenarios, for instance when one matrix is a scalar multiple of the other or again when they are simultaneously diagonalizable. Simonedl (talk) 18:24, 13 October 2023 (UTC)

 * The article is correct. The initial statement says that the multiples of the identity are the only matrices which commute with everything, not the only ones that commute with some things.  I.e., that these are exactly the elements of the center. 35.139.154.158 (talk) 19:47, 13 October 2023 (UTC)
 * Ah, now it is clear. Since English is not my native language, I misunderstood the sentence. Thank you for the clarification. Simonedl (talk) 23:51, 13 October 2023 (UTC)
 * Note that your first citation requires commutation of the given A with every B. So you are not free to chose B = c I, nor to restrict B to a diagonal matrix, nor to require B to be a scalar multiple of A or to be simultaneously diagonalizable with A. - Jochen Burghardt (talk) 21:25, 13 October 2023 (UTC)

title: Talk:Matrix multiplication


text:
Kanuri language

Kanuri is a Saharan dialect continuum of the Nilo–Saharan language family spoken by the Kanuri and Kanembu peoples in Nigeria, Niger, Chad and Cameroon, as well as by a diaspora community residing in Sudan.

Background
At the turn of the 21st century, its two main dialects, Manga Kanuri and Yerwa Kanuri (also called Beriberi, which its speakers consider to be pejorative), were spoken by 9,700,000 people in Central Africa. It belongs to the Western Saharan subphylum of Nilo-Saharan. Kanuri is the language associated with the Kanem and Bornu empires that dominated the Lake Chad region for a thousand years.

The basic word order of Kanuri sentences is subject–object–verb. It is typologically unusual in simultaneously having postpositions and post-nominal modifiers – for example, 'Bintu's pot' would be expressed as nje Bintu-be, 'pot Bintu-of'.

Kanuri has three tones: high, low, and falling. It has an extensive system of consonantal lenition; for example, sa- 'they' + -buma 'have eaten' → za-wuna 'they have eaten'.

Traditionally a local lingua franca, its usage has declined in recent decades. Most first-language speakers speak Hausa or Arabic as a second language.

Geographic distribution
Kanuri is spoken mainly in lowlands of the Chad Basin, with speakers in Cameroon, Chad, Niger, Nigeria, Sudan and Libya.

Nigeria
The Kanuri region in Nigeria consists of Borno State and Yobe State. Some other states such as Jigawa, Gombe and Bauchi are also dominated by Kanuri people, but they are not included in this region. Cities and towns where Kanuri is spoken include Maiduguri, Damaturu, Hadejia, Akko, Duku, Kwami, Kano, Kaduna, Gusau, Jos and Lafia.

In central Nigeria, the Kanuri are usually referred to as Bare-Bari or Beriberi.

Central Kanuri, also known as Yerwa Kanuri, is the main language of the Kanuri people living in Borno State, Yobe State and Gombe State, and it is usually referred to as Kanuri in Nigeria.

Manga Kanuri, which is the main language of the Kanuri people in Yobe State, Jigawa State and Bauchi State, is usually referred to as Manga or Mangari or Mangawa, and they are distinct from the Kanuri, which is a term generally used for speakers of Central Kanuri.

The Kanembu language is also spoken in Borno State on the border with Chad.

Niger
In Niger, the Kanuri region is composed of Diffa Region and Zinder Region in the southeast. Parts of Agadez Region are also Kanuri. Cities where it is spoken include Zinder, Diffa, N'Guigmi and Bilma.

In Zinder region, the main dialect is Manga. In Diffa Region, the main dialect is Tumari or Kanembu; Kanembu is spoken by a minority. In Agadez Region, the main dialect is Bilma. Central Kanuri is a minority dialect, and is commonly referred to as Bare-Bari or Beriberi.

Varieties
 Ethnologue divides Kanuri into the following languages, while many linguists (e.g. Cyffer 1998) regard them as dialects of a single language. The first three are spoken by ethnic Kanuri and thought by them as dialects of their language.

 * Central Kanuri
 * Manga Kanuri
 * Tumari Kanuri
 * Bilma Kanuri (or Bilms)
 * Kanembu

The variety attested in 17th-century Qur'anic glosses is known as Old Kanembu. In the context of religious recitation and commentaries, a heavily archaizing descendant of this is still used, called Tarjumo.

Consonants

 * There may also exist prenasalized voiced stop consonant sounds, although it is not known whether they are considered phonemic.
 * The sound occurs mainly as an allophone of, when following another voiceless plosive. It also may be in free alteration with ; however, it is still represented in the standard Kanuri orthography.
 * A voiceless fricative occurs as an allophone of  when preceding back vowels . A voiced fricative  occurs as an allophone of, when occurring in intervocalic positions. A voiced fricative  occurs as an allophone of , when occurring intervocalically between central vowels.
 * A retroflex lateral sound is heard when  is followed by.
 * occurs as an allophone of when preceding velar stop consonants. Often, the stop sounds are deleted or misheard, so just the nasal sound  is mainly heard.

Vowels

 * is heard as an allophone of.

Written Kanuri
Kanuri has been written using the Ajami Arabic script, mainly in religious or court contexts, for at least four hundred years. More recently, it is also sometimes written in a modified Latin script. The Gospel of John published in 1965 was produced in Roman and Arabic script.

Alphabet
A standardized romanized orthography (known as the Standard Kanuri Orthography in Nigeria) was developed by the Kanuri Research Unit and the Kanuri Language Board. Its elaboration, based on the dialect of Maiduguri, was carried out by the Orthography Committee of the Kanuri Language Board, under the Chairmanship of Abba Sadiq, Waziri of Borno. It was officially approved by the Kanuri Language Board in Maiduguri, Nigeria, in 1975.

Letters used : a b c d e ǝ f g h i j k l m n ny o p r ɍ s sh t u w y z.

Sample text in Kanuri (Universal Declaration of Human Rights)
''Hakkiwa-a nambe a suro Wowur abəden dəganadə ndu-a nduana-aso kartaa, gayirtə futubibema baaro, gayirta alama jiilibeso, kadigəbeso, alagəbeso, təlambeso, adinbeso, siyasabeso au rayiwu, lardə gade au kaduwu gade, kənganti, tambo au awowa laa gade anyiga samunzəna. Anyibe ngawoman nduma kəla siyasaben, kal kəntəwoben kal au daraja dunyalabe sawawuro kal kərye au lardə kamdə dəganabe sawawuro gayirtinba. Lardə shi gultənama adə kərmai kəlanzəben karga, amanaro musko lardə gadeben karga, kəlanzəlangənyi karga au sədiya kaidawa kəntəwobe laan karga yaye kal.''

Translation

Everyone is entitled to all the rights and freedoms set forth in this Declaration, without distinction of any kind, such as race, colour, sex, language, religion, political or other opinion, national or social origin, property, birth or other status. Furthermore, no distinction shall be made on the basis of the political, jurisdictional or international status of the country or territory to which a person belongs, whether it be independent, trust, non-self-governing or under and other limitation of sovereingty.

(Article 1 of the Universal Declaration of Human Rights).

title: Kanuri language


text: Talk:List of metropolitan areas by population


text:
Talk:Czechoslovakia

Bias
The bias at the page is visible via particular usage of terms. I suggest using neutral ones because Wikipedia is an independent encyclopedia and such bias damages neutrality. --Comrade-yutyo (talk) 07:57, 29 September 2020 (UTC)
 * "Bias"? that’s all you have to say to justify your reverts? GizzyCatBella  🍁  08:12, 29 September 2020 (UTC)
 * User talk:GizzyCatBella my edits contain neutral terms, while yours contain strawman. You do the same thing at every article about the particular political view. --Comrade-yutyo (talk) 08:23, 29 September 2020 (UTC)
 * Yuto is right, GizzyCatBella is pushing hysterical non-neutral language. GizzyCatBella is also denying the role of the Slovak state in the Holocaust of the Jews by removing "and their allies" from the Holocaust sentence. The Slovak state deported the Jews to their death, not the Nazis, The Holocaust in Slovakia.--Iron Thain (talk) 09:18, 29 September 2020 (UTC)
 * Sorry the page will be restored status quo ante, arguments needed first for a wise discussion.(KIENGIR (talk) 16:22, 30 September 2020 (UTC))
 * Can you show the same neutrality for articles related to socialism? The way of language used in particular articles regarding socialist Poland, Hungary and Czechoslovakia is so biased. --Comrade-yutyo (talk) 07:36, 1 October 2020 (UTC)
 * Please point by point explain your concerns.(KIENGIR (talk) 12:27, 2 October 2020 (UTC))

"ŘČS" listed at Redirects for discussion
An editor has identified a potential problem with the redirect ŘČS and has thus listed it for discussion. This discussion will occur at Redirects for discussion/Log/2022 March 13 until a consensus is reached, and readers of this page are welcome to contribute to the discussion. – filelakeshoe (t / c) 🐱 14:49, 13 March 2022 (UTC)

untitled
I would like to ask why is here presented as official coat of arms of Czechoslovakia the coat of arms of Czechoslovak socialist republic. I do not deny that socialist coat of arms was used between 1960 and 1990. But this period was not the most defining period of Czechoslovakia. In fact, socialist regime hijacked the Czechoslovak state. The founding idea of Czechoslovakia was not soviet communism. The founding idea of Czechoslovakia came from Tomáš Garrigue Masaryk, which was quite opposite of soviet communism. If Czechoslovakia was founded as state based on soviet communism, it would make sense, if there is presented the socialist coat of arms. But the soviet communism rather hijacked Czechoslovak state and replaced its original coat of arms by the socialist one. It would make sense, if you replaced the communist coat of arms either by coat of arms of Czechoslovakia right before its dissolution or by coat of arms of Czechoslovakia based on idea of Masaryk, i. e. the coat of arms that was official between 1920 and 1960. — Preceding unsigned comment added by 2A02:830A:B20D:D300:F9C7:6C6F:3532:60DA (talk) 12:33, 9 July 2022 (UTC)

"Czech -" listed at Redirects for discussion
An editor has identified a potential problem with the redirect Czech - and has thus listed it for discussion. This discussion will occur at Redirects for discussion/Log/2022 December 3 until a consensus is reached, and readers of this page are welcome to contribute to the discussion. Steel1943 (talk) 21:53, 3 December 2022 (UTC)

Article contradicts itself
Their are two dates used for the dissoloution of the state, being December 31, 1992 and January 1, 1993 and they are used interchangeably. Blackmamba31248 (talk) 00:47, 7 May 2024 (UTC)

title: Talk:Czechoslovakia


text:
Bunker Hill

Bunker Hill, Bunkers Hill or Bunker's Hill may refer to:

Massachusetts, U.S.

 * Bunker Hill, after which the Battle of Bunker Hill was named, a hill in the Boston neighborhood of Charlestown
 * Battle of Bunker Hill, a 1775 American Revolutionary War battle fought near the hill
 * USS Bunker Hill (CV-17), an Essex-class aircraft carrier
 * USS Bunker Hill (CG-52), a Ticonderoga-class cruiser
 * Bunker Hill Community College, a college in Charlestown, Boston

Other places in the U.S.

 * Bunker Hill, Los Angeles, California, a district in downtown Los Angeles
 * Bunker Hill, Illinois
 * Bunker Hill Township, Macoupin County, Illinois
 * Bunker Hill, Indiana, a town in Miami County
 * Grissom Joint Air Reserve Base or Bunker Hill Air Force Base
 * Bunker Hill, Fayette County, Indiana
 * Bunker Hill, Morgan County, Indiana
 * Bunker Hill, Kansas
 * Bunker Hill (Millersville, Maryland)
 * Bunker Hill Township, Michigan
 * Bunker Hill, Howard County, Missouri
 * Bunker Hill, Lewis County, Missouri
 * Bunker Hill, Stoddard County, Missouri
 * Bunker Hill (Nevada), a mountain in the Toiyabe Range
 * Bunker Hill, Oregon
 * Bunker Hill, Tennessee
 * Bunker Hill, Washington
 * Bunker Hill, West Virginia
 * Bunker Hill, Wisconsin

Places in Australia

 * Bunkers Hill, Victoria, location near Ballarat, Australia

Places in Jamaica

 * Bunkers Hill, Jamaica, location in Trelawny Parish

Places in England

 * Bunkers Hill, Cambridgeshire, a location in England
 * Bunkers Hill, Greater Manchester, a location in England
 * Bunkers Hill, Lincolnshire, a location in England
 * Bunkers Hill, Norfolk, a location in England
 * Bunker's Hill, Nottingham, location of St Stephen's Church, Bunker's Hill
 * Bunkers Hill, Oxfordshire, a location in EnglandK
 * Bunkers Hill, Suffolk, a location in England
 * Bunker's Hill, Wolverhampton, an area of Wolverhampton, England

Other uses

 * Bunker Hill (film), a film by Kevin Willmott
 * Bunker Hill (musician), American R&B and gospel singer
 * Bunker Hill (song), a 2003 song by the Red Hot Chili Peppers
 * Bunker Hill Historic District
 * "Bunker Hill" (Supergirl), an episode of Supergirl
 * The Death of General Warren at the Battle of Bunker's Hill, June 17, 1775, painting

title: Bunker Hill