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Andrew Sullivan

Andrew Michael Sullivan (born 10 August 1963) is a British-American author, editor, and blogger. Sullivan is a political commentator, a former editor of The New Republic, and the author or editor of six books. He started a political blog, The Daily Dish, in 2000, and eventually moved his blog to platforms, including Time, The Atlantic, The Daily Beast, and finally an independent subscription-based format. He announced his retirement from blogging in 2015. From 2016 to 2020, Sullivan was a writer-at-large at New York. He launched his newsletter The Weekly Dish in July 2020.

Sullivan has stated that his conservatism is rooted in his Catholic background and in the ideas of the British political philosopher Michael Oakeshott. In 2003, he wrote that he could no longer support the American conservative movement, as he was disaffected with the Republican Party's continued rightward shift towards social conservatism on social issues during the George W. Bush era.

Born and raised in Britain, he has lived in the United States since 1984. He is openly gay and a practicing Catholic.

Early life and education
Sullivan was born in South Godstone, Surrey, England, into a Catholic family of Irish descent, and was brought up in the nearby town of East Grinstead, West Sussex. He was educated at a Catholic primary school followed by Reigate Grammar School, where his classmates included future Prime Minister of the United Kingdom Keir Starmer who is also the leader of the Labour Party and future Conservative member of the House of Lords Andrew Cooper. He won a scholarship in 1981 to Magdalen College, Oxford, where he was awarded a first-class Bachelor of Arts in modern history and modern languages. He founded the Pooh Stick Society at Oxford, and in his second year, he was elected President of the Oxford Union for Trinity term 1983.

After writing briefly for a newspaper, Sullivan won a scholarship in 1984 to Harvard University, where he earned a Master of Public Administration in 1986 from the John F. Kennedy School of Government, followed by a Doctor of Philosophy degree in government from Harvard in 1990. His dissertation was titled Intimations Pursued: The Voice of Practice in the Conversation of Michael Oakeshott.

Career
Sullivan first wrote for The Daily Telegraph on American politics. In 1986, Sullivan went to work for The New Republic magazine initially on a summer internship; among the most significant articles he wrote were "Gay Life Gay Death", an essay on the AIDS crisis, and "Sleeping with the Enemy", in which he attacked the practice of "outing", both of which earned him some recognition among the gay community. He was appointed the editor of The New Republic in October 1991, a position he held until 1996. In that position, he expanded the magazine from its traditional roots in political coverage to cultural issues and the politics surrounding them. During this time, the magazine generated several high-profile controversies.

While completing graduate work at Harvard in 1988, Sullivan published an attack in Spy magazine on Rhodes Scholars, "All Rhodes Lead Nowhere in Particular," which dismissed recipients of the scholarship as "hustling apple-polisher[s]"; "high-profile losers"; "the very best of the second-rate"; and "misfits by the very virtue of their bland, eugenic perfection." "[T]he sad truth is that as a rule," Sullivan wrote, "Rhodies possess none of the charms of the aristocracy and all of the debilities: fecklessness, excessive concern that peasants be aware of their achievement, and a certain hemophilia of character." Author Thomas Schaeper notes that "[i]ronically, Sullivan had first gone to the United States on a Harkness Fellowship, one of many scholarships spawned in emulation of the Rhodes program."

In 1994, Sullivan published excerpts on race and intelligence from Richard Herrnstein and Charles Murray's controversial The Bell Curve, which argued that some of the measured difference in IQ scores among racially defined groups was a result of genetic inheritance. Almost the entire editorial staff of the magazine threatened to resign if material that they considered racist was published. To appease them, Sullivan included lengthy rebuttals from 19 writers and contributors. He has continued to speak approvingly of the research and arguments presented in The Bell Curve, writing, "The book ... still holds up as one of the most insightful and careful of the last decade. The fact of human inequality and the subtle and complex differences between various manifestations of being human—gay, straight, male, female, black, Asian—is a subject worth exploring, period." According to Sullivan, this incident was a turning point in his relationship with the magazine's staff and management, which he conceded was already bad because he "was a lousy manager of people." He left the magazine in 1996.

Sullivan began writing for The New York Times Magazine in 1998, but editor Adam Moss fired him in 2002. Jack Shafer wrote in Slate magazine that he had asked Moss in an e-mail to explain this decision, but that his e-mails went unanswered, adding that Sullivan was not fully forthcoming on the subject. Sullivan wrote on his blog that the decision had been made by Times executive editor Howell Raines, who found Sullivan's presence "uncomfortable," but defended Raines's right to fire him. Sullivan suggested that Raines did so in response to Sullivan's criticism of the Times on his blog, and said he had expected that his criticisms would eventually anger Raines.

Sullivan has also worked as a columnist for The Sunday Times of London.

Ross Douthat and Tyler Cowen have suggested that Sullivan is the most influential political writer of his generation, particularly because of his very early and strident support for same-sex marriage, his early political blog, his support of the Iraq War, and his subsequent support of Barack Obama's presidential candidacy.

After the cessation of his long-running blog, The Dish, in 2015, Sullivan wrote regularly for New York during the 2016 presidential election, and in February 2017 he began writing a weekly column, "Interesting Times", for the magazine.

On July 19, 2020, following the unexplained absence of his column for June 5, Sullivan announced that he would no longer write for New York. He announced he would be reviving The Dish as a newsletter, The Weekly Dish, hosted by Substack.

Politics
Sullivan describes himself as a conservative and is the author of The Conservative Soul. He has supported a number of traditional libertarian positions, favouring limited government and opposing social interventionist measures such as affirmative action. However, on a number of controversial public issues, including same-sex marriage, social security, progressive taxation, anti-discrimination laws, the Affordable Care Act, the United States government's use of torture, and capital punishment, he has taken positions not typically shared by conservatives in the United States. In July 2012, Sullivan said that "the catastrophe of the Bush–Cheney years ... all but exploded the logic of neoconservatism and its domestic partner-in-crime, supply-side economics."

One of the most important intellectual and political influences on Sullivan is Michael Oakeshott. Sullivan describes Oakeshott's thought as "an anti-ideology, a nonprogramme, a way of looking at the world whose most perfect expression might be called inactivism." He argues "that Oakeshott requires us to systematically discard programmes and ideologies and view each new situation sui generis. Change should only ever be incremental and evolutionary. Oakeshott viewed society as resembling language: it is learned gradually and without us really realising it, and it evolves unconsciously, and for ever." In 1984, he wrote that Oakeshott offered "a conservatism which ends by affirming a radical liberalism." This "anti-ideology" is perhaps the source of accusations that Sullivan "flip-flops" or changes his opinions to suit the whims of the moment. He has written, "A true conservative—who is, above all, an anti-ideologue—will often be attacked for alleged inconsistency, for changing positions, for promising change but not a radical break with the past, for pursuing two objectives—like liberty and authority, or change and continuity—that seem to all ideologues as completely contradictory."

As a youth, Sullivan was a fervent supporter of Margaret Thatcher and later Ronald Reagan. He says of that time, "What really made me a right-winger was seeing the left use the state to impose egalitarianism—on my school", after the Labour government in Britain tried to merge his admissions-selective school with the local comprehensive school. At Oxford, he became friends with future prominent conservatives William Hague and Niall Ferguson and became involved with Conservative Party politics.

From 1980 through 2000, Sullivan supported Republican presidential candidates in the United States, with the exception of the election of 1992, when he supported Bill Clinton in his first presidential campaign. In 2004, however, he was angered by George W. Bush's support of the Federal Marriage Amendment designed to enshrine in the Constitution marriage as a union between a man and a woman, as well as what he saw as the Bush administration's incompetence over its Iraq War management, and consequently supported the presidential campaign of John Kerry, a Democrat.

Sullivan endorsed Senator Barack Obama for the Democratic nomination in the 2008 United States presidential election, and Representative Ron Paul for the Republican nomination. After John McCain clinched the Republican primary and named Sarah Palin as his vice presidential nominee, Sullivan began to espouse a birther-like conspiracy theory involving Palin and her young son Trig. Sullivan devoted a significant amount of space in The Atlantic, questioning whether Palin is Trig's biological mother. He and others who held this belief, dubbed "Trig Truthers", demanded Palin produce a birth certificate or other piece of medical evidence to prove Trig is indeed Palin's biological son.

Sullivan eventually endorsed Obama for president, largely because he believed that he would restore "the rule of law and Constitutional balance"; he also argued that Obama represented a more realistic prospect for "bringing America back to fiscal reason", and expressed a hope that Obama would be able to "get us past the culture war." Sullivan continued to maintain that Obama was the best choice for president from a conservative point of view. During the 2012 election campaign, he wrote, "Against a radical right, reckless, populist insurgency, Obama is the conservative option, dealing with emergent problems with pragmatic calm and modest innovation. He seeks as a good Oakeshottian would to reform the country's policies in order to regain the country's past virtues. What could possibly be more conservative than that?" Sullivan has declared support for Arnold Schwarzenegger and other like-minded Republicans. He argues that the Republican Party, and much of the conservative movement in the United States, has largely abandoned its earlier scepticism and moderation in favour of a more fundamentalist certainty, both in religious and political terms. He has said this is the primary source of his alienation from the modern Republican Party.

In January 2009, Forbes ranked Sullivan No. 19 on a list of "The 25 Most Influential Liberals in the U.S. Media". Sullivan rejected the "liberal" label and set out his grounds in a published article in response.

In August 2018, after Sarah Jeong, an editorial board-member of The New York Times, received widespread criticism for her old anti-white tweets, Sullivan accused Jeong of being racist and calling white people "subhuman". Sullivan also accused Jeong of spreading eliminationist rhetoric; a belief that political opponents are a societal cancer that should be separated, censored or exterminated.

HIV
In 1996, discussing HIV, he argued in the New York Times Magazine that "this plague is over" insofar as "it no longer signifies death. It merely signifies illness." This led to "a trend of white male journalists proclaiming that AIDS is over", according to Sarah Schulman.

Gay issues
Sullivan, like Marshall Kirk, Hunter Madsen, and Bruce Bawer, has been described by Urvashi Vaid as a proponent of "legitimation", seeing the objective of the gay rights movement as being "mainstreaming gay and lesbian people" rather than "radical social change". Sullivan wrote the first major article in the United States advocating for gay people to be given the right to marry, published in The New Republic in 1989. According to one columnist for Intelligent Life, many on "the gay left," aiming to alter social codes of sexuality for everyone, were chagrined at Sullivan's endorsement of the "assimilation" of gay people into "straight culture." In the wake of the United States Supreme Court rulings on same-sex marriage in 2013 (Hollingsworth v. Perry and United States v. Windsor), The New York Times op-ed columnist Ross Douthat suggested that Sullivan might be the most influential political writer of his generation, writing: "No intellectual that I can think of, writing on a fraught and controversial topic, has seen their once-crankish, outlandish-seeming idea become the conventional wisdom so quickly, and be instantiated so rapidly in law and custom."

As of 2007, Sullivan opposed hate crime laws, arguing that they undermine freedom of speech and equal protection.

In 2014, Sullivan opposed calls to remove Brendan Eich as CEO of Mozilla for donating to the campaign for Proposition 8, which made same-sex marriage illegal in California. In 2015, he claimed that "gay equality" had been achieved in the United States through the persuasive arguments of "old-fashioned liberalism" rather than by the activism of "identity politics leftism."

Transgender issues
In 2007, he said he was "no big supporter" of the Employment Non-Discrimination Act, arguing that it would "not make much of a difference." He said the "gay rights establishment" was making a tactical error to insist on including protections for gender identity, however, as he believed it would be easier to pass the bill without transgender people.

In a September 2019 Intelligencer column, Sullivan expressed concern that gender-nonconforming children (especially those who are likely one day to come out as gay) might be encouraged to believe that they are transgender when they are not. In November 2019, Sullivan wrote another Intelligencer column on young women who, in their teens, had begun to transition to live as men but who later detransitioned. In that article, he discussed the controversy over a 2018 journal article by Lisa Littman that proposed a socially mediated subtype of gender dysphoria that Littman had termed "rapid onset gender dysphoria". In April 2021, he said it should be illegal for doctors to initiate cross-sex hormones for children under 16 or sex reassignment surgery for children under 18.

Recognitions
In 1996, Sullivan's book, Virtually Normal: An Argument about Homosexuality, won the 1996 Mencken Award for Best Book, presented by the Free Press Association. In 2006, Sullivan was named an LGBT History Month icon.

Iraq war, war on terror
Sullivan supported the 2003 invasion of Iraq by the United States and was initially hawkish in the war on terror, arguing that weakness would embolden terrorists. He was "one of the most militant" supporters of the Bush administration's counter-terrorism strategy immediately following the September 11 attacks in 2001; in an essay for The Sunday Times, he stated, "The middle part of the country—the great red zone that voted for Bush—is clearly ready for war. The decadent Left in its enclaves on the coasts is not dead—and may well mount what amounts to a fifth column." Eric Alterman wrote in 2002 that Sullivan had "set himself up as a one-man House Un-American Activities Committee" running an "inquisition" to unmask "anti-war Democrats", "basing his argument less on the words these politicians speak than on the thoughts he knows them to be holding in secret".

Later, Sullivan criticised the Bush administration for its prosecution of the war, especially regarding the numbers of troops, protection of munitions, and treatment of prisoners, including the use of torture against detainees in United States custody. Though he argued that enemy combatants in the war on terror should not have been given status as prisoners of war because "terrorists are not soldiers", he believed that the US government was required to abide by the rules of war—in particular, Article 3 of the Geneva Conventions—when dealing with such detainees. In retrospect, Sullivan said that the torture and abuse of prisoners at the Abu Ghraib prison in Iraq had jolted him back to "sanity". Of his early support for the invasion of Iraq, he said, "I was terribly wrong. In the shock and trauma of 9/11, I forgot the principles of scepticism and doubt towards utopian schemes that I had learned."

On the edition of 27 October 2006 of Real Time with Bill Maher, he described conservatives and Republicans who refused to admit they had been wrong to support the Iraq War as "cowards". On 26 February 2008, he wrote on his blog: "After 9/11, I was clearly blinded by fear of al Qaeda and deluded by the overwhelming military superiority of the US and the ease of democratic transitions in Eastern Europe into thinking we could simply fight our way to victory against Islamist terror. I wasn't alone. But I was surely wrong." His reversal on the Iraq issue and his increasing attacks on the Bush administration caused a severe backlash from many hawkish conservatives, who accused him of not being a "real" conservative.

Sullivan authored an opinion piece, "Dear President Bush," that was featured as the cover article of the October 2009 edition of The Atlantic. In it, he called on Bush to take personal responsibility for the incidents and practices of torture that occurred during his administration as part of the war on terror.

Israel
Sullivan states that he has "always been a Zionist". However, his views of Israel have become more critical over time. In February 2009, he wrote that he could no longer take the neoconservative position on Israel seriously.

In January 2010, Sullivan blogged that he was "moving toward" the idea of "a direct American military imposition" of a two-state solution to the Israeli-Palestinian conflict, with NATO troops enforcing "the borders of the new states of Palestine and Israel". He commented, "I too am sick of the Israelis. [...] I'm sick of having a great power like the US being dictated to." His post was criticised by Noah Pollak of Commentary, who referred to it as "crazy", "heady stuff" based on "hubris".

In February 2010, Leon Wieseltier suggested in The New Republic that Sullivan, a former friend and colleague, had a "venomous hostility toward Israel and Jews" and was "either a bigot, or just moronically insensitive" toward the Jewish people. Sullivan rejected the accusation and was defended by some writers, while others at least partly supported Wieseltier.

In March 2019, Sullivan wrote in New York magazine that while he strongly supported the right of a Jewish state to exist, he felt that U.S. Representative Ilhan Omar's comments about the influence of the pro-Israel lobby were largely correct. Sullivan said that "it is simply a fact that the Israel lobby uses money, passion, and persuasion to warp this country's foreign policy in favor of another country — out of all proportion to what Israel can do for the US."

Iran
Sullivan devoted a significant amount of blog space to covering the allegations of fraud and related protests after the 2009 Iranian presidential election. Francis Wilkinson of The Week stated that Sullivan's "coverage—and that journalism term takes on new meaning here—of the uprising in Iran was nothing short of extraordinary. 'Revolutionary' might be a better word."

Sullivan was inspired by the Iranian people's reactions to the election results and used his blog as a hub of information. Because of the media blackout in Iran, Iranian Twitter accounts were a major source of information. Sullivan frequently quoted and linked to Nico Pitney of The Huffington Post.

Immigration
Writing for New York magazine, Sullivan expressed concern that high levels of immigration to the United States could drive "white anxiety" by causing white Americans to be "increasingly troubled by the pace of change" since they were never formally asked whether they wanted such a demographic shift. Sullivan has advocated for tighter immigration controls on asylum and overall lower levels of immigration. He has criticized Democrats for what he perceived as their unwillingness to implement such controls.

Race
As editor at The New Republic, Sullivan published excerpts from the 1994 book The Bell Curve, by Richard Herrnstein and Charles Murray. The book, which contained a chapter about the subject of IQ on society and public policy, argued that there are innate differences in intelligence between racial groups. This view of an innate connection between race and intelligence is rejected by the majority of scientists.

In 2015 Jeet Heer, in an article in The New Republic entitled "The New Republic's Legacy on Race" described Sullivan's decision as an example of "The magazine's myopia on racial issues". The importance of Sullivan to the popularization of The Bell Curve and race science was noted by Matthew Yglesias who called Sullivan "the punditocracy's original champion of Murray's thinking on genetics". Similarly, Gavin Evans wrote in The Guardian that Sullivan "was one of the loudest cheerleaders for The Bell Curve in 1994" and that he "returned to the fray in 2011, using his popular blog, The Dish, to promote the view that population groups had different innate potentials when it came to intelligence."

Blogging
In late 2000, Sullivan began his blog, The Daily Dish. The core principle of the blog has been the style of conservatism he views as traditional. This includes fiscal conservatism, limited government, and classic libertarianism on social issues. Sullivan opposes government involvement with respect to sexual and consensual matters between adults, such as the use of marijuana and prostitution. He believes recognition of same-sex marriage is a civil-rights issue but expressed willingness to promote it on a state-by-state legislative federalism basis, rather than trying to judicially impose the change. Most of Sullivan's disputes with other conservatives have been over social issues and the handling of postwar Iraq.

Sullivan gave out yearly "awards" for various public statements, parodying those of the people the awards were named after. Throughout the year, nominees were mentioned in various blog posts. The readers of his blog chose winners at the end of each year.
 * The Hugh Hewitt Award, introduced in June 2008 and named after a man Sullivan described as an "absurd partisan fanatic", was for the most egregious attempts to label Barack Obama as un-American, alien, treasonous, and far out of the mainstream of American life and politics.
 * The John Derbyshire Award was for egregious and outlandish comments on gays, women, and minorities.
 * The Paul Begala Award was for extreme liberal hyperbole.
 * The Michelle Malkin Award was for shrill, hyperbolic, divisive, and intemperate right-wing rhetoric. (Ann Coulter was ineligible for this award so that, in Sullivan's words, "other people will have a chance.")
 * The Michael Moore Award was for divisive, bitter, and intemperate left-wing rhetoric.
 * The Matthew Yglesias Award was for writers, politicians, columnists, or pundits who criticised their own side of the political spectrum, made enemies among political allies, and generally risked something for the sake of saying what they believed.
 * The "Poseur Alert" was awarded for passages of prose that stood out for pretension, vanity, and bad writing designed to look profound.
 * The Dick Morris Award (formerly the Von Hoffman Award) was for stunningly wrong cultural, political, and social predictions. Sullivan renamed this award in September 2012, saying that Von Hoffman was "someone who in many ways got the future right—at least righter than I did."

In February 2007, Sullivan moved his blog from Time to The Atlantic Monthly, where he had accepted an editorial post. His presence was estimated to have contributed as much as 30% of the subsequent traffic increase for The Atlantic website.

In 2009, The Daily Dish won the 2008 Weblog Award for Best Blog.

Sullivan left The Atlantic to begin blogging at The Daily Beast in April 2011. In 2013, he announced that he was leaving The Daily Beast to launch The Dish as a stand-alone website, charging subscribers $20 a year.

In a note posted on The Dish on 28 January 2015, Sullivan announced his decision to retire from blogging. He posted his final blog entry on 6 February 2015. On 26 June 2015, he posted an additional piece in reaction to Obergefell v. Hodges, which legalized same-sex marriage in the United States.

In July 2020, Sullivan announced that The Dish would be revived as a weekly feature, including a column and podcast; he published there and elsewhere a notable obituary of Queen Elizabeth II.

Personal life
In 2001, it came to light that Sullivan had posted anonymous online advertisements for unprotected anal sex, preferably with "other HIV-positive men". He was widely criticised in the media for this, with some critics noting that he had condemned President Bill Clinton's "incautious behavior", though others wrote in his defence.

In 2003, Sullivan wrote a Salon article identifying himself as a member of the gay "bear community". On 27 August 2007, he married Aaron Tone in Provincetown, Massachusetts. On May 26, 2023, Sullivan announced on his blog that he and Tone had gotten divorced the previous week.

Sullivan was barred for many years from applying for United States citizenship because of his HIV-positive status. Following the statutory and administrative repeals of the HIV immigration ban in 2008 and 2009, respectively, he announced his intention to begin the process of becoming a permanent resident and citizen. On The Chris Matthews Show on 16 April 2011, Sullivan confirmed that he had become a permanent resident, showing his green card. On 1 December 2016, Sullivan became a naturalised US citizen.

He has been a daily user of marijuana since 2001.

Religion
Sullivan identifies himself as a faithful Catholic while disagreeing with some aspects of the Catholic Church's doctrine.

He expressed concern about the election of Pope Benedict XVI in a Time magazine article on 24 April 2005, titled "The Vicar of Orthodoxy". He wrote that Benedict was opposed to the modern world and women's rights, and considered gays and lesbians innately disposed to evil. Sullivan has, however, agreed with Benedict's assertion that reason is an integral element of faith.

Sullivan takes a moderate approach to religion, rejecting fundamentalism and describing himself as a "dogged defender of pluralism and secularism." Sullivan was a friend of late journalist and atheist writer Christopher Hitchens, and often debated religion with him. Sullivan also defended religious moderates in a series of exchanges with atheist author Sam Harris.

Works

 * As author
 * Virtually Normal: An Argument About Homosexuality (1995). Knopf. ISBN 0-679-42382-6.
 * Love Undetectable: Notes on Friendship, Sex and Survival (1998). Knopf. ISBN 0-679-45119-6.
 * The Conservative Soul: How We Lost It, How to Get It Back (2006). HarperCollins. ISBN 0-06-018877-4.
 * Intimations Pursued: The Voice of Practice in the Conversation of Michael Oakeshott (2007). Imprint Academic. ISBN 978-0-907845-28-7
 * Out on a Limb: Selected Writing, 1989–2021 (2021). Avid Reader Press. ISBN 978-1501155895
 * As editor
 * Same-Sex Marriage Pro & Con: A Reader (1997). Vintage. ISBN 0-679-77637-0. First edition
 * Same-Sex Marriage Pro & Con: A Reader (2004). Vintage. ISBN 1-4000-7866-0. Second edition
 * The View from Your Window: The World as Seen by Readers of One Blog (2009). Blurb.com

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

title: Andrew Sullivan


text:
Talk:Trace (linear algebra)

orthogonal
{1 0}

{0 1}

and

{1/sqrt(2) -1/sqrt(2)}

{1/sqrt(2) 1/sqrt(2)}

Both are orthonormal bases. The diagonals aren't equal. Is the definition of trace seriously just summing along the diagonal?

 * Yes it is. Seriously. You seem to make some unwarranted assumption and I can only guess what it is that you are doing wrong. Idea 1: You assume that orthogonal matrices all have the same trace. This is not true. Udea 2: You seem to assume that similar (conjugate) matrices have the same trace (that is true). But the two matrices you have here are not similar. The one realizes the identical function the other realizes a rotation. — Preceding unsigned comment added by 217.95.169.8 (talk) 14:49, 13 January 2019 (UTC)

— Preceding unsigned comment added by 107.3.37.113 (talk) 07:49, 1 August 2015 (UTC)

I do not think that the article title "Trace (matrix)" is any clearer than "Trace of a matrix". In fact, quite the opposite.

Wikipedia articles are about concepts, not about words. This is not about the word "trace" in the context of matrix theory (in which case it should properly be called "Trace (matrix theory)" or "Trace (linear algebra)"), but it is about the concept "Trace of a matrix", which is a perfectly self-explanatory headline of the article. AxelBoldt 23:10 Jan 31, 2003 (UTC)

 * Trace is empty. let's move it there -- Tarquin 23:26 Jan 31, 2003 (UTC)

I would like to see 2 things: Brief paragraph on the importace of the trace, and an intoative proof of the main feature (is it main?) tr(AB) = tr(BA). thanks
 * I agree that this article should be renamed. Mainly because the concept of the trace is more general that that of a matrix.  In other words there is a class of linear transformations for which the trace makes sense which are not matrices.  move the article to Trace -Lethe | Talk

Coordinate-free definition
Why is the map given by $$v \otimes h \mapsto (w \mapsto h(w)v)$$? If I'm not mistaken this would suggest that the image of any linear operator is one-dimensional. 99.240.144.168 (talk) 02:23, 31 December 2011 (UTC)

Trace of matrix products
I would like to suggest a change regarding the trace of the product of several (>2) matrices. Specifically the part beginning "If A, B, and C are square...". The phrase "all permutations" is, strictly speaking, not correct. The last three terms in the chained equality, while looking like general permutations, are transposes of the first three with the transpose marks omitted due to symmetry. My motivation for the change is that if more than three matrices are involved only the cyclic permutation may be used to reorder the matrices. Transposing with symmetry assumed will make more (apparent) permutations possible, but not all posible permutations. My quick and dirty suggestion is to delete the entire paragraph along with the chained equality. It could be fixed but it is just a special case involving the previously discussed cyclic permutation and transpose along with the symmetry condition. Doggit 17:19, 10 August 2006 (UTC)

Direct Link
I set the link from spur to this article. --Kajaktiger 16:47, 4 November 2006 (UTC)

Derivatives
Don't revert back to d Tr = I. This is not true. There is a vectorisation operator present if you must present the information in this form. The peterson & pederson seem to have dropped this when they copy pasted from brookes.

Just goes to show what sort of crap you can find on the internet. —The preceding unsigned comment was added by 150.203.45.188 (talk) 09:35, 5 December 2006 (UTC).

Inner product
I think that following $$ = tr\left(A\,^*B\right)$$, it should be mentioned that $$^*B$$ is the conjugate transpose of B. 131.130.90.152 10:36, 22 January 2007 (UTC)

Trace of a 1×1 matrix
Trace can be used in a non-obvious way where one considers a scalar the trace of a 1×1 matrix. For example, if X is a n×1 column vector and A is a n×n square matrix, then $$X^TAX$$ is a 1×1 matrix which is often regarded as a scalar, but it can be beneficial to use the trace operator instead (to use the cyclic property). This technique is used in the Estimation of covariance matrices article. Can anyone provide a simpler example to add to the main article? Roman V. Odaisky 17:23, 4 October 2007 (UTC)

$$\mathrm{tr}(AB)^n$$
I’m confused by the formula $$0 \leq \mathrm{tr}(AB)^n \leq \mathrm{tr}(A)^n \mathrm{tr}(B)^n$$. Shouldn’t it have read $$\mathrm{tr}((AB)^n)$$ instead of the middle term? Or perhaps the formula should be split into two inequalities:

 * $$\mathrm{tr} AB \leq \mathrm{tr} A \cdot \mathrm{tr} B$$
 * $$\mathrm{tr} A^n \leq (\mathrm{tr} A)^n$$ (if n is natural then this follows from the previous inequality)

Roman V. Odaisky 17:33, 4 October 2007 (UTC)

Various edits
Hi---just wanted to leave a note about the various edits I've been making to the article. If anyone has objections or comments, please respond here---thanks. Spireguy (talk) 18:28, 4 March 2008 (UTC)

Intuition/example
I know the determinate of a transformation matrix can be thought of as the change in volume it causes. For example, if det(A) = 1, then A doesn't change the (n-dimensional) volume of an object it is applied to. Could someone provide a similar intuition for trace? What is the meaning of the sum of the eigenvalues? —Ben FrantzDale (talk) 11:59, 27 March 2008 (UTC)

This is addressed in the article under "derivatives", although perhaps it is not clear as stated. The trace measures the infinitesimal change in volume, since it is the derivative of the determinant. Do you think this should be emphasized or re-stated in a more down-to-earth manner? -- Spireguy (talk) 16:41, 27 March 2008 (UTC)

OK, I went ahead and put in an example. Comments? -- Spireguy (talk) 16:49, 27 March 2008 (UTC)

Symmetric and anti-symmetric matrices
Let $$A$$ be a symmetric matrix, and $$B$$ anti-symmetric matrix. THen
 * $$\operatorname{tr}(AB) = 0$$

I was going to add this to the article, but can't see the proof on top of my head. -- Taku (talk) 02:40, 1 October 2008 (UTC)

Proof: $$ A = A^T \quad B = -B^T $$

So

$$ (AB)^T = -BA$$

$$\operatorname{tr}(AB)=\operatorname{tr}( (AB)^T ) = - \operatorname{tr}( BA) = - \operatorname{tr}(AB)$$

And thus

$$ \Rightarrow \operatorname{tr}(AB)=0$$

wpoely86 (talk) 16:58, 24 February 2010 (UTC)

the trace of a matrix is the sum of its eigenvalues?
Is this statement true?? Doesn't that only apply for hermitian matrixes which have been diagonalized?? 20:11, 17 January 2009 (UTC) —Preceding unsigned comment added by 128.113.65.174 (talk)

No, it always holds, even for non-diagonalizable matrices. As the article notes, you can see Jordan normal form for a further explanation. -- Spireguy (talk) 02:26, 18 January 2009 (UTC)

I find this statement confusing too. Wouldn't be better to formulate it as "The trace of a matrix is equal to the sum of the (complex) eigenvalues,..." 31 October 2012 — Preceding unsigned comment added by Fuzzyrandom (talk • contribs) 13:20, 31 October 2012 (UTC)

why is the reference section being used for footnotes?
????? —Preceding unsigned comment added by 86.20.235.138 (talk) 12:47, 14 March 2009 (UTC)

Trace of a commutator
The article contains the text "When both A and B are n by n, the trace of the commutator of A and B vanishes: tr([A, B]) = 0. Conversely, any square matrix with zero trace is the commutator of some two matrices.[3] In particular, the identity matrix is never similar to the commutator of any matrices.". Surely this can't be true! I mean, if you take two nxn matrices which commute, then their trace will equal the trace of the identity matrix, which is never zero. Is this meant to say something else? SetaLyas (talk) 12:53, 25 May 2009 (UTC)

I think you are confusing two different notions of "commutator." The relevant one here is [A,B]=AB-BA, whereas you seem to be thinking of A^(-1)B^(-1)AB. When A,B commute, the former is zero, while the latter is the identity matrix. The latter version is not directly relevant to the trace. -- Spireguy (talk) 13:26, 25 May 2009 (UTC)

Algebraic Multiplicities
"If A is a square n-by-n matrix with real or complex entries and if λ1,...,λn are the (complex and distinct) eigenvalues of A (listed according to their algebraic multiplicities)"

The line says that A is n-by-n and that the eigenvalues \lambda_1...\lambda_n should be distinct. Then it requires the eigenvalues to be listed accordingly to their algebraic multiplicity. But if the eigenvalues are all distinct, of course their algebraic multiplicities are 1. So in what the (listed according ...) helps? —Preceding unsigned comment added by 79.138.132.212 (talk) 08:41, 4 September 2009 (UTC)

$$\mathrm{tr}(AB)^n$$
Can somebody point me to a proof of the statement: $$0 \leq \mathrm{tr}(AB)^n \leq \mathrm{tr}(A)^n \mathrm{tr}(B)^n$$?
 * No, since it's totally false. Algebraist 22:06, 24 February 2010 (UTC)
 * I've found a proof for the case of n=1 for a real positive semidefinit matrix, so for general n it's also proven. wpoely86 (talk) 11:24, 25 February 2010 (UTC)

Similarity Invariance
Perhaps it should be mentioned that the matrix P must have the same dimensions as A. --Tomtheebomb (talk) 22:46, 3 April 2010 (UTC)

Also this might be linked in here: Von Neumann's trace inequality http://en.wikipedia.org/wiki/Von_Neumann%27s_trace_inequality --Tomtheebomb (talk) 23:34, 3 April 2010 (UTC)

Trace only for diagonal matrix?
The first line of the article says "the trace of an n-by-n diagonal matrix A is defined to be the sum of the elements ...", whereas, the first example shows calculation of trace for a non-diagonal matrix. As far as I know, trace is defined for square matrix, and does not require diagonal matrix. -- Samikrc (talk) 05:30, 7 February 2011 (UTC)

Trace of a symmetric matrix 0?
How can it be zero always? It is mentioned under the section example. — Preceding unsigned comment added by Tagib (talk • contribs) 07:03, 14 December 2011 (UTC)

 * Because it is false. Any simple example can prove that. However, the trace of a product of a symmetric and skew-symmetric matrix is always zero. wpoely86 (talk) 13:22, 3 October 2012 (UTC)

Trace of a projection matrix
The trace of a projection matrix is the dimension of the target space. $$P_{X}=X\left(X^{T}X\right)^{-1}X$$, we have $$\text{Tr}\left(P_{X}\right)=\text{rank}\left(X\right)$$. Jackzhp (talk) 03:08, 9 February 2011 (UTC)

"Trace is cognate to 'spoor'"
This doesn't make any sense. Can we clarify this point? —Preceding unsigned comment added by 152.3.68.8 (talk) 23:12, 23 March 2011 (UTC)

Paratrace?
Is a sum of the elements on a diagonal parallel to the primary diagonal given a name?

At least one use of $$\operatorname{tr_p}$$ is familiar. If $$D$$ is a distance matrix, a permutation of indices of $$D$$ which minimizes $$\operatorname{tr_1}(D)$$ is a solution of the traveling salesman problem. Regards, ... PeterEasthope (talk) 02:52, 7 October 2019 (UTC)

Coordinate free definition (2)
The claim: "the space of linear operators on a finite-dimensional vector space $V$ (defined over the field $F$) is isomorphic to the space $V ⊗ V^{∗}$ is not correct: the space of linear operators on $V$ has dimension n^2 while the space $V ⊗ V^{∗}$ has dimension 2n. Clearly that map between the two spaces is not surjective.
 * You are incorrect. The space $V ⊗ V^{∗}$ has dimension n^2. Gumshoe2 (talk) 02:36, 27 January 2022 (UTC)

"Coordinate-free definition" is not coordinate-free
It is ok to say that the map $$V\otimes V^\ast\to F$$ is defined in a coordinate-free way, but the isomorphism of $$V\otimes V^\ast$$ with $$Hom(V,V)$$ cannot be understood without coordinates/bases. So the section is saying that coordinates/bases are not used, but it is just using a linguistic trick of avoiding explicitly pointing out the part where they are used. The essential content of the section is fine but it should be phrased as just putting the trace in the language of tensors, not as giving a definition that doesn't use coordinates. (It also may be more appropriate on the wiki page for tensor product.) Gumshoe2 (talk) 02:45, 27 January 2022 (UTC)

 * I thought the isomorphism $$V\otimes V^*\rightarrow \text{Hom}(V,V)$$ was the linear extension of
 * $$u\otimes \varphi \mapsto A$$ where $$A(v) = \varphi(v)u.$$
 * This direction at least looks coordinate independent to me, though I'm not sure how to construct the inverse in a coordinate-free way. But as long as this is an isomorphism, then we can just define the inverse as being the inverse to this map, and that is coordinate-free.
 * To show the map is an isomorphism, since the dimension of both spaces is the same we need only show the kernel is trivial. It's not too hard to show this. Zephyr the west wind (talk) 10:37, 11 July 2022 (UTC)

Cyclic property of trace: why 4 matrices instead of 3?
In the article, cyclicity of trace is shown by the equation

Why are 4 matrices used here as opposed to 3 (say just A, B, C)? Using three matrices would demonstrate perfectly well that trace is invariant under cycling. Zephyr the west wind (talk) 10:43, 11 July 2022 (UTC)

Trace of AB and BA matrices
The statement "It can also be proven that tr(AB) = tr(BA) for any two matrices A and B." is incorrect. This equality holds only for matrices with compatible structure so that both AB and BA are defined. If A is an m x n matrix, B is an n x p matrix and m /= p, the BA product is not defined, and the trace is not defined for AB either.

The statement can be rectified by either

a) stating that A and B are m x n and n x m matrices respectively (as below, but this is confusing, it took me a while to notice this special structure in the text)

b) equivalently, stating that AB product is a square matrix (then BA is also square)

c) stating that "tr(AB) = tr(BA) for any two matrices A and B for which the trace is defined." (implying (b)).

d) Using the "Frobenius product" article notation: "tr(ATB) = tr(BAT) for any two m x n matrices A and B"

Which would be the most clear? Pterodaktilis (talk) 07:01, 22 July 2023 (UTC)

title: Talk:Trace (linear algebra)


text:
Talk:Neolithic architecture

Untitled
This is a rather strange article which contradicts itself, beginning by stating that neolithic "architecture" pre-dates written history, then going on to speak of written eyewitness accounts. I don't want to be dismissive, but the last paragraph seems more or less irrelevant. I propose to delete it unless the author would like to explain a bit more clearly what the intention of the article is, and demonstrate that it doesn't infringe the NPOV rule. Deb 23:11 Jan 31, 2003 (UTC)

What is the NPOV rule?Kar 13:36 Feb 1, 2003 (UTC)

Hello, Kar. Nice to meet you. The NPOV rule means that all articles must keep to a "neutral point of view" -- in other words, they should be limited to facts and not personal opinions. That doesn't mean that you can't include ideas you personally believe in, as long as you are careful how you phrase them. For example, in this article (which I'm assuming you wrote), you refer to the Old Testament as a historical document. I think a lot of people would dispute that it is historically accurate and would describe it as a collection of fictional stories used by ancient people to explain things they didn't understand. However, if you believe that it is true, you can still say that "Many people believe that the Old Testament is a historical document" and go on to cite the evidence for this. I think what you've already written is very interesting, but it will undoubtedly be altered - not necessarily by me, as I am a Christian and respect fundamentalist beliefs which I don't share, but by people who don't accept what you say. So you need to be careful how you say it. Deb 17:40 Feb 1, 2003 (UTC)

End of Neolithic level in the Americas
Regarding this statement: "The people of the Americas and the Pacific remained at the Neolithic level of technology up until the time of European contact, including very small exceptions (a few copper hatchets and spear heads in the Great Lakes region).[citation needed]"

I might be misunderstanding it, my English is pretty basic. But I think is not right, according to this:

http://en.wikipedia.org/wiki/Metallurgy_in_pre-Columbian_America

Indigenous South Americans had full metallurgy with smelting and various metals being purposely alloyed. Some of them might be considered Bronze age technologies. — Preceding unsigned comment added by 70.28.245.98 (talk) 17:27, 5 May 2014 (UTC)

title: Talk:Neolithic architecture


text:
Talk:Conventicle Act 1664

Untitled
What is all this "17 Charles II c. 2" stuff? It makes it sound like a reference to a verse in the bible or sumfink. -- SGBailey 23:32 Jan 31, 2003 (UTC)

 * It means get the statute book for the 17th year of the reign of King Charles II, and turn to chapter 2. This is a normal way of referencing Acts of Parliament from before 1962. --rbrwr

 * One slightly intriguing and possibly confusing thing about Charles II, specifically, is that his reign is postdated to include the whole Protectorate. You cannot figure when a statute was passed by counting forward from the coronation. --Ihcoyc

Date of repeal
According to the Chronological Table of the Statutes (HMSO), this Act was repealed by the Statute Law Revision Act 1863. James500 (talk) 17:27, 25 November 2008 (UTC)

Doesn't Explain Provisions of the Act
The article doesn't explain the actual provisions of the act. What's up with that? 162.196.229.52 (talk) 00:51, 21 May 2015 (UTC)

Antecedents
It has its antecedents in the Religion Act 1592 and other Acts of the era that were mostly about increased surveillance of potential separatists. Should these antecedents be mentioned in the article? — Preceding unsigned comment added by 50.176.228.45 (talk) 19:24, 6 November 2016 (UTC)

A Commons file used on this page or its Wikidata item has been nominated for deletion
The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for deletion: Participate in the deletion discussion at the nomination page. —Community Tech bot (talk) 17:41, 30 January 2023 (UTC)
 * Coat of Arms of England (1660-1689).svg

title: Talk:Conventicle Act 1664


text:
Talk:Italian War of Independence

Untitled
It is surprising that in the account of the Second War of Independence, no reference is made ro Garibaldi and his Expedition of the Thousand. This was the key event that enabled Cavour to become the first Prime Miistet md Victor Emmanuel to be the first King of Italy. I casn write more about this another time. From David Lewsi davedlewis@yahoo.com Jan 31 2003

This is garbage
This needs to be reworked considerably. It was probably written by an Italian.

Justa joking (but not about the reworking part). —The preceding unsigned comment was added by 154.5.23.99 (talk) 03:40, 22 March 2007 (UTC).

World War I
I have added an internal link to World War I, there should be a section likening the war to a fourth unification/independence war for Italy, as it is viewed by many as the final realization of a united Italian state. --NEMT 02:09, 5 June 2007 (UTC)

Good -- I have seen war memorials in Italy refering to World War One as the "Fourth Independence War". --Amcalabrese (talk) 17:30, 18 June 2008 (UTC)

title: Talk:Italian War of Independence


text:
User:Bbjeff

"Everyone is able to change everyting?"

Read about this idea in a german computer magazine and liked it.

One of my favourites is Michael Jordan, so for sure I looked for entrys about him first. Already changed something, not really knowing about all the format standards etc., but it is a start.

I know my grammar is terrible, but I'll try my best. I hope I will improve.

title: User:Bbjeff


text:
User:CrusadeOnTerrorists

CrusadeOnTerrorists chose to change hir username to ShameOnTerrorists, after being informed that some people might find the word "crusade" in that context inappropriate.

Welcome to Wikipedia. Please read and understand Neutral point of view before adding material. Tannin

title: User:CrusadeOnTerrorists


text:
Ourang Medan

The SS Ourang Medan was a supposed ghost ship which, according to various sources, became a shipwreck in the Dutch East Indies (modern Indonesia) in the Straits of Malacca waters, or elsewhere, after its entire crew had died under suspicious circumstances, either in 1940, 1947 or 1948, depending on the newspaper source. Some argue that the story of the Ourang Medan is an urban legend due to the fact that there were never records of a ship named the Ourang Medan.

The Mystery of the SS Ourang Medan
One English reference to the ship and the incident was published in the May 1952 issue of the Proceedings of the Merchant Marine Council, published by the United States Coast Guard. An earlier English reference was published on October 10, 1948, in The Albany Times of Albany, New York and references its original source as Elsevier's Weekly. The word Ourang (also written Orang) is Malay or Indonesian for "man" or "person", whereas Medan is the largest city on the Indonesian island of Sumatra, giving an approximate translation of "Man of Medan". Accounts of the ship's accident have appeared in various books and magazines, mainly on Forteana. Their factual accuracy and even the ship's existence, however, are unconfirmed, and details of the vessel's construction and history, if any, remain unknown. Searches for any official registration or accident investigation recorded have proven unsuccessful.

The story's first appearance was a series of three articles in the Dutch-Indonesian newspaper De locomotief: Samarangsch handels- en advertentie-blad (February 3, 1948, with two photographs, February 28, 1948, and March 13, 1948). The name of the ship that found the Ourang Medan is never mentioned, but the location of the encounter is described as 400 nmi southeast of the Marshall Islands. The second and third articles describe the experiences of the sole survivor of the Ourang Medan crew, who was found by an Italian missionary and natives on Taongi Atoll in the Marshall Islands. The man, before perishing, tells the missionary that the ship was carrying a badly stowed cargo of oil of vitriol, and that most of the crew perished because of the poisonous fumes escaping from broken containers. According to the story, the Ourang Medan was sailing from an unnamed small Chinese port to Costa Rica, and deliberately avoided the authorities. The survivor, an unnamed German, died after telling his story to the missionary, who told the story to the author, Silvio Scherli of Trieste, Italy. The Dutch newspaper concludes with a disclaimer: "'This is the last part of our story about the mystery of the Ourang Medan. We must repeat that we don't have any other data on this 'mystery of the sea'. Nor can we answer the many unanswered questions in the story. It may seem obvious that the entire story is a fantasy, a thrilling romance of the sea. On the other hand, the author, Silvio Scherli, assures us of the authenticity of the story.'" Silvio Scherli is said to have produced a report on Trieste "Export Trade" on September 28, 1959.

New evidence found by The Skittish Library shows there were 1940 newspaper reports of the incident taken from the Associated Press in British newspapers the Daily Mirror and the Yorkshire Evening Post. Again, there were differences in the story: the location being the Solomon Islands, and the SOS messages different from later reports. The story still appears to originate with Silvio Scherli in Trieste.

Circumstances
According to the story, at some point of time in or around June 1947 (Gaddis and others list the approximate date as early February 1948 ), two American vessels navigating the Straits of Malacca, the City of Baltimore and the Silver Star, among others passing by, picked up several distress messages from the nearby Dutch merchant ship Ourang Medan. A radio operator aboard the troubled vessel sent the following message in Morse code: "S.O.S. from Ourang Medan * * * We float. All officers including the captain, dead in chartroom and on the bridge. Probably whole of crew dead * * *." After a few more incoherent dots and dashes, the words "I die." were received. After this message, there was nothing more heard of. When the Silver Star crew eventually located and boarded the apparently-undamaged Ourang Medan in an attempt at a rescue, the ship was found littered with corpses (including the carcass of a dog) everywhere, with the dead bodies found sprawled on their backs, the frozen (and allegedly badly-frightened) faces of the deceased upturned to the sun above with mouths gaping open and eyes staring straight ahead, with the corpses resembling horrible caricatures. No survivors were located and no visible signs of injuries on the dead bodies were observed. Just as the ship was to be prepared for a tow by the Silver Star to a nearby port, a fire then suddenly broke out in the ship's No. 4 cargo-hold, forcing the boarding party to hastily evacuate the doomed Dutch freighter, thus preventing any further investigations to be carried out. Soon after, the Ourang Medan was witnessed exploding before finally sinking.

Unsecured hazardous materials cargo
Bainton and others hypothesize that Ourang Medan might have been involved in smuggling operations of chemical substances such as a combination of potassium cyanide and nitroglycerin or even wartime stocks of nerve agents. According to these theories, sea water would have entered the ship's hold, reacting with the cargo to release toxic gases, which then caused the crew to succumb to asphyxia and/or poisoning. Later, the sea water would have reacted with the nitroglycerin, causing the reported fire and explosion.

Another theory is that the ship was transporting nerve gas which the Japanese military had been storing in China during the war, and which was handed over to the U.S. military at the end of the war. No U.S. ship could transport it as it would leave a paper trail. It was therefore loaded onto a non-registered ship for transport to the U.S. or an island in the Pacific.

Carbon monoxide (CO) poisoning
Gaddis puts forward the theory that an undetected smouldering fire or malfunction in the ship's boiler system might have been responsible for the shipwreck. Escaping carbon monoxide would have caused the deaths of all aboard, with the fire slowly spreading out of control, leading to the vessel's ultimate destruction.

Skepticism
Several authors note their inability to find any mention of the case in Lloyd's Shipping Register. Furthermore, no registration records for a ship by the name of Ourang Medan could be located in various countries, including the Netherlands. While author Roy Bainton states that the identity of the Silver Star, reported to have been involved in the failed rescue attempt, has been established with high probability, the complete lack of information on the sunken ship itself has given rise to suspicion about the origins and credibility of the account. Ships logs for the Silver Star did not show a record of any such rescue attempt. Bainton and others have put forward the possibility that accounts of, among others, the date, location, names of the ships involved, and circumstances of the accident might have been inaccurate or exaggerated, or that the story might be completely fictitious.

One British researcher has found the story of the Ourang Medan, transposed to the Solomon Islands, but also with a Trieste connection, in two British newspapers in 1940 (The Yorkshire Evening Post for 21 November 1940 and The Daily Mirror for 22 November 1940 ), both quoting AP (The Associated Press) news agency.

title: Ourang Medan