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Diagonal matrix

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal matrix is $$\left[\begin{smallmatrix} 3 & 0 \\ 0 & 2 \end{smallmatrix}\right]$$, while an example of a 3×3 diagonal matrix is$$ \left[\begin{smallmatrix} 6 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 4 \end{smallmatrix}\right]$$. An identity matrix of any size, or any multiple of it is a diagonal matrix called a scalar matrix, for example, $$\left[\begin{smallmatrix} 0.5 & 0 \\ 0 & 0.5 \end{smallmatrix}\right]$$. In geometry, a diagonal matrix may be used as a scaling matrix, since matrix multiplication with it results in changing scale (size) and possibly also shape; only a scalar matrix results in uniform change in scale.

Definition
As stated above, a diagonal matrix is a matrix in which all off-diagonal entries are zero. That is, the matrix $D = (d_{i,j})$ with $n$ columns and $n$ rows is diagonal if $$\forall i,j \in \{1, 2, \ldots, n\}, i \ne j \implies d_{i,j} = 0.$$

However, the main diagonal entries are unrestricted.

The term diagonal matrix may sometimes refer to a , which is an $m$-by-$n$ matrix with all the entries not of the form $d_{i,i}$ being zero. For example: $$\begin{bmatrix} 1 & 0 & 0\\ 0 & 4 & 0\\  0 & 0 & -3\\  0 & 0 & 0\\ \end{bmatrix} \quad \text{or} \quad \begin{bmatrix} 1 & 0 & 0 & 0 & 0\\ 0 & 4 & 0& 0 & 0\\  0 & 0 & -3& 0 & 0 \end{bmatrix}$$

More often, however, diagonal matrix refers to square matrices, which can be specified explicitly as a . A square diagonal matrix is a symmetric matrix, so this can also be called a .

The following matrix is square diagonal matrix: $$\begin{bmatrix} 1 & 0 & 0\\ 0 & 4 & 0\\ 0 & 0 & -2 \end{bmatrix}$$

If the entries are real numbers or complex numbers, then it is a normal matrix as well.

In the remainder of this article we will consider only square diagonal matrices, and refer to them simply as "diagonal matrices".

Vector-to-matrix diag operator
A diagonal matrix $$\mathbf{D}$$ can be constructed from a vector $$\mathbf{a} = \begin{bmatrix}a_1 & \dotsm & a_n\end{bmatrix}^\textsf{T}$$ using the $$\operatorname{diag}$$ operator: $$\mathbf{D} = \operatorname{diag}(a_1, \dots, a_n)$$

This may be written more compactly as $$\mathbf{D} = \operatorname{diag}(\mathbf{a})$$.

The same operator is also used to represent block diagonal matrices as $$ \mathbf{A} = \operatorname{diag}(A_1, \dots, A_n)$$ where each argument $$A_i$$ is a matrix.

The $$\operatorname{diag}$$ operator may be written as: $$\operatorname{diag}(\mathbf{a}) = \left(\mathbf{a} \mathbf{1}^\textsf{T}\right) \circ \mathbf{I}$$ where $$\circ$$ represents the Hadamard product and $$\mathbf{1}$$ is a constant vector with elements 1.

Matrix-to-vector diag operator
The inverse matrix-to-vector $$\operatorname{diag}$$ operator is sometimes denoted by the identically named $$\operatorname{diag}(\mathbf{D}) = \begin{bmatrix}a_1 & \dotsm & a_n\end{bmatrix}^\textsf{T}$$ where the argument is now a matrix and the result is a vector of its diagonal entries.

The following property holds: $$\operatorname{diag}(\mathbf{A}\mathbf{B}) = \sum_j \left(\mathbf{A} \circ \mathbf{B}^\textsf{T}\right)_{ij} = \left( \mathbf{A} \circ \mathbf{B}^\textsf{T} \right) \mathbf{1} $$

Scalar matrix
A diagonal matrix with equal diagonal entries is a scalar matrix; that is, a scalar multiple λ of the identity matrix $I$. Its effect on a vector is scalar multiplication by λ. For example, a 3×3 scalar matrix has the form: $$ \begin{bmatrix} \lambda &      0 & 0       \\ 0 & \lambda & 0      \\ 0 &      0 & \lambda \end{bmatrix} \equiv \lambda \boldsymbol{I}_3 $$

The scalar matrices are the center of the algebra of matrices: that is, they are precisely the matrices that commute with all other square matrices of the same size. By contrast, over a field (like the real numbers), a diagonal matrix with all diagonal elements distinct only commutes with diagonal matrices (its centralizer is the set of diagonal matrices). That is because if a diagonal matrix $$\mathbf{D} = \operatorname{diag}(a_1, \dots, a_n)$$ has $$a_i \neq a_j,$$ then given a matrix $$\mathbf{M}$$ with $$m_{ij} \neq 0,$$ the $$(i, j)$$ term of the products are: $$(\mathbf{D}\mathbf{M})_{ij} = a_im_{ij}$$ and $$(\mathbf{M}\mathbf{D})_{ij} = m_{ij}a_j,$$ and $$a_jm_{ij} \neq m_{ij}a_i$$ (since one can divide by $$m_{ij}$$), so they do not commute unless the off-diagonal terms are zero. Diagonal matrices where the diagonal entries are not all equal or all distinct have centralizers intermediate between the whole space and only diagonal matrices.

For an abstract vector space V (rather than the concrete vector space $$K^n$$), the analog of scalar matrices are scalar transformations. This is true more generally for a module M over a ring R, with the endomorphism algebra End(M) (algebra of linear operators on M) replacing the algebra of matrices. Formally, scalar multiplication is a linear map, inducing a map $$R \to \operatorname{End}(M),$$ (from a scalar λ to its corresponding scalar transformation, multiplication by λ) exhibiting End(M) as a R-algebra. For vector spaces, the scalar transforms are exactly the center of the endomorphism algebra, and, similarly, scalar invertible transforms are the center of the general linear group GL(V). The former is more generally true free modules $$M \cong R^n$$, for which the endomorphism algebra is isomorphic to a matrix algebra.

Vector operations
Multiplying a vector by a diagonal matrix multiplies each of the terms by the corresponding diagonal entry. Given a diagonal matrix $$\mathbf{D} = \operatorname{diag}(a_1, \dots, a_n)$$ and a vector $$\mathbf{v} = \begin{bmatrix} x_1 & \dotsm & x_n \end{bmatrix}^\textsf{T}$$, the product is: $$\mathbf{D}\mathbf{v} = \operatorname{diag}(a_1, \dots, a_n)\begin{bmatrix}x_1 \\ \vdots \\ x_n\end{bmatrix} = \begin{bmatrix} a_1 \\ & \ddots \\ &       & a_n \end{bmatrix} \begin{bmatrix}x_1 \\ \vdots \\ x_n\end{bmatrix} = \begin{bmatrix}a_1 x_1 \\ \vdots \\ a_n x_n\end{bmatrix}. $$

This can be expressed more compactly by using a vector instead of a diagonal matrix, $$\mathbf{d} = \begin{bmatrix} a_1 & \dotsm & a_n \end{bmatrix}^\textsf{T}$$, and taking the Hadamard product of the vectors (entrywise product), denoted $$\mathbf{d} \circ \mathbf{v}$$:

$$\mathbf{D}\mathbf{v} = \mathbf{d} \circ \mathbf{v} = \begin{bmatrix} a_1 \\ \vdots \\ a_n \end{bmatrix} \circ \begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix} = \begin{bmatrix} a_1 x_1 \\ \vdots \\ a_n x_n \end{bmatrix}. $$

This is mathematically equivalent, but avoids storing all the zero terms of this sparse matrix. This product is thus used in machine learning, such as computing products of derivatives in backpropagation or multiplying IDF weights in TF-IDF, since some BLAS frameworks, which multiply matrices efficiently, do not include Hadamard product capability directly.

Matrix operations
The operations of matrix addition and matrix multiplication are especially simple for diagonal matrices. Write $diag(a_{1}, ..., a_{n})$ for a diagonal matrix whose diagonal entries starting in the upper left corner are $a_{1}, ..., a_{n}$. Then, for addition, we have

$$ \operatorname{diag}(a_1,\, \ldots,\, a_n) + \operatorname{diag}(b_1,\, \ldots,\, b_n) = \operatorname{diag}(a_1 + b_1,\, \ldots,\, a_n + b_n)$$

and for matrix multiplication,

$$\operatorname{diag}(a_1,\, \ldots,\, a_n) \operatorname{diag}(b_1,\, \ldots,\, b_n) = \operatorname{diag}(a_1 b_1,\, \ldots,\, a_n b_n).$$

The diagonal matrix $diag(a_{1}, ..., a_{n})$ is invertible if and only if the entries $a_{1}, ..., a_{n}$ are all nonzero. In this case, we have

$$\operatorname{diag}(a_1,\, \ldots,\, a_n)^{-1} = \operatorname{diag}(a_1^{-1},\, \ldots,\, a_n^{-1}).$$

In particular, the diagonal matrices form a subring of the ring of all $n$-by-$n$ matrices.

Multiplying an $n$-by-$n$ matrix $A$ from the left with $diag(a_{1}, ..., a_{n})$ amounts to multiplying the $i$-th row of $A$ by $a_{i}$ for all $i$; multiplying the matrix $A$ from the right with $diag(a_{1}, ..., a_{n})$ amounts to multiplying the $i$-th column of $A$ by $a_{i}$ for all $i$.

Operator matrix in eigenbasis
As explained in determining coefficients of operator matrix, there is a special basis, $e_{1}, ..., e_{n}$, for which the matrix $$\mathbf{A}$$ takes the diagonal form. Hence, in the defining equation $\mathbf{A} \mathbf e_j = \sum_i a_{i,j} \mathbf e_i$, all coefficients $$a_{i,j} $$ with $i ≠ j$ are zero, leaving only one term per sum. The surviving diagonal elements, $$a_{i,i}$$, are known as eigenvalues and designated with $$\lambda_i$$ in the equation, which reduces to $$\mathbf{A} \mathbf e_i = \lambda_i \mathbf e_i$$. The resulting equation is known as eigenvalue equation and used to derive the characteristic polynomial and, further, eigenvalues and eigenvectors.

In other words, the eigenvalues of $diag(λ_{1}, ..., λ_{n})$ are $λ_{1}, ..., λ_{n}$ with associated eigenvectors of $e_{1}, ..., e_{n}$.

Properties

 * The determinant of $diag(a_{1}, ..., a_{n})$ is the product $a_{1}⋯a_{n}$.
 * The adjugate of a diagonal matrix is again diagonal.
 * Where all matrices are square,
 * A matrix is diagonal if and only if it is triangular and normal.
 * A matrix is diagonal if and only if it is both upper- and lower-triangular.
 * A diagonal matrix is symmetric.
 * The identity matrix $I_{n}$ and zero matrix are diagonal.
 * A 1×1 matrix is always diagonal.
 * The square of a 2×2 matrix with zero trace is always diagonal.

Applications
Diagonal matrices occur in many areas of linear algebra. Because of the simple description of the matrix operation and eigenvalues/eigenvectors given above, it is typically desirable to represent a given matrix or linear map by a diagonal matrix.

In fact, a given $n$-by-$n$ matrix $A$ is similar to a diagonal matrix (meaning that there is a matrix $X$ such that $X^{−1}AX$ is diagonal) if and only if it has $n$ linearly independent eigenvectors. Such matrices are said to be diagonalizable.

Over the field of real or complex numbers, more is true. The spectral theorem says that every normal matrix is unitarily similar to a diagonal matrix (if $AA^{∗} = A^{∗}A$ then there exists a unitary matrix $U$ such that $UAU^{∗}$ is diagonal). Furthermore, the singular value decomposition implies that for any matrix $A$, there exist unitary matrices $U$ and $V$ such that $U^{∗}AV$ is diagonal with positive entries.

Operator theory
In operator theory, particularly the study of PDEs, operators are particularly easy to understand and PDEs easy to solve if the operator is diagonal with respect to the basis with which one is working; this corresponds to a separable partial differential equation. Therefore, a key technique to understanding operators is a change of coordinates—in the language of operators, an integral transform—which changes the basis to an eigenbasis of eigenfunctions: which makes the equation separable. An important example of this is the Fourier transform, which diagonalizes constant coefficient differentiation operators (or more generally translation invariant operators), such as the Laplacian operator, say, in the heat equation.

Especially easy are multiplication operators, which are defined as multiplication by (the values of) a fixed function–the values of the function at each point correspond to the diagonal entries of a matrix.

title: Diagonal matrix


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SDF

SDF may refer to:

File formats

 * Simple Data Format, for binary data
 * Spatial Data File, for geodatabases
 * Standard Delay Format, for timing data
 * SQL Server Compact Edition Database File (filename extension: )
 * Structure data file, for chemical tables
 * Scientific Data Format, a Hierarchical Data Format implementation

Formal computer science

 * Signed distance function (or field), in mathematical applications
 * Syntax Definition Formalism, to describe formal languages

Other uses in computing

 * Software development folder, a physical or virtual container for software project artifacts
 * Synchronous Data Flow, a restriction of Kahn process networks
 * SDF Public Access Unix System, a shared shell provider

Entertainment and media

 * Südtirol Digital Fernsehen, a TV station in South Tyrol, Italy
 * Super Dimensional Fortress, warships in the Robotech/Macross franchise
 * Settlement Defense Front (SDF), an enemy faction of Call of Duty: Infinite Warfare

Military forces

 * Japan Self-Defense Forces
 * State defense force, US
 * Sudan Defence Force, 1925–1955
 * Syrian Democratic Forces, North and East Syria

Political parties

 * Sikkim Democratic Front, India
 * Social Democratic Federation (United States), 1936–1956
 * Social Democratic Federation, UK, 1884–1911
 * Social Democratic Forum, Hong Kong, 2000
 * Social Democratic Front (disambiguation), Cameroon and Ghana
 * Socialist Democratic Federation (Japan), 1978–1994

Other organizations

 * Scouts de France, youth group
 * Serb Democratic Forum, Serbs of Croatia

Places

 * Louisville International Airport, Kentucky, US (by IATA code)
 * Soquel Demonstration State Forest, California, US

Science

 * Silver diammine fluoride, in dentistry
 * Stromal cell-derived factor (disambiguation), SDF1, SDF1a, etc.
 * Subwavelength-diameter optical fibre
 * Synchronous diaphragmatic flutter or hiccup

Other uses

 * SDF Group (SAME Deutz-Fahr), an Italian-based agricultural machine manufacturer
 * Simplified directional facility, an aviation instrument approach navigational aid
 * Stochastic discount factor, in econometrics

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

title: SDF


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Samuel O'Reilly

Samuel F. O’Reilly (May 1854 - 29 April 1909) was an American tattoo artist from New York, who patented the first electric tattoo machine on December 8, 1891.

Biography
O’Reilly was born in Waterbury, New Haven County, Connecticut, to Irish immigrants Thomas O’Reilly and Mary Ann Hurley in May 1854.

He began tattooing in New York around the mid-1880s, probably mentored by Martin Hildebrandt. O'Reilly's machine was based on the rotary technology of Thomas Edison's autographic printing pen. Although O'Reilly held the first patent for an electric tattoo machine, tattoo artists had been experimenting with and modifying a variety of different machines prior to the issuance of the patent. O'Reilly's first pre-patent tattoo machine was a modified dental plugger, which he used to tattoo several dime museum attractions for exhibition between the years 1889 and 1891. From the late 1880s on, tattoo machines continually evolved into the modern tattoo machine.

O'Reilly first owned a shop at #5 Chatham Square on the New York Bowery. In 1904, he moved to #11 Chatham Square when the previous tenant, tattoo artist Elmer Getchell, left the city. Charles Wagner was allegedly apprenticed to O'Reilly and later assumed ownership of his #11 Chatham Square shop.

On April 29, 1909, Samuel O'Reilly fell while painting his house and died. He is buried in Holy Cross Cemetery, Brooklyn, NY.

title: Samuel O'Reilly


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Talk:Crème brûlée

Spelling
This should be crème. Or US write crême to make it look more French ?

Both spellings: "crème" and "crême", are present in the Wikipedia index for this dessert. I do not know which is more common in the U.S. Perhaps, since the Creole people speak French, the latter is preferred.JSF 15:18 Jan 22, 2003 (UTC)

 * English speaking writers will often leave out French accents when writing French names 'creme' (without an accent) occurs frequently but Crème brûlée is french french, not an anclicised word and professional food writers would not omit the accents as a rule. It looks sloppy if you are not consistent with the accents in a printed publication. The web is not a printed publication, so don't depend on it as an authority on spelling.Mike Hayes (talk) 18:21, 14 June 2008 (UTC)

 * "Crème" is the French spelling. (some of the stray ^s might well be due to me. *kicks self*) -- Tarquin 15:00 Jan 22, 2003 (UTC)

 * Delia Smith (veteran TV cook in UK) spells it crème brûlée (note the circumflex on the u), but the Oxford Dictionary for Writers and Editors (OUP, 2000) spells it without the circumflex. OED2 records crème as the main spelling with crême as an alternative, and shows examples of brulée with and without a circumflex.  So take your pick.  -- Heron

 * Correct French spelling (Petit Larousse) is crème (accent grave). - Montréalais

 * Can we all agree that the spelling is Crème brûlée? The Anome

 * The Dictionnaire de l'Académie française, neuvième édition spells it crème brûlée.Mike Hayes (talk) 18:02, 14 June 2008 (UTC)

 * It is in dispute. You should have moved the talk. Now one will have to be deleted. Mintguy

 * The previous talk has now been moved here (see above). The Anome 00:36 Jan 25, 2003 (UTC)

 * Well, brûlée is completely wrong French. Google don't make it right. I protest at yet another move. -- Tarquin 00:24 Jan 25, 2003 (UTC)

 * Google test:
 * site:fr brûlée = 10,800
 * site:fr brulée = 1040

What other options are there. i'm confused now. Mintguy

 * brûlée looks 100% correct to me, native french speaker. Of course I'm not infallible, but the nearest dictionary (which happens to be a French-English one) thinks like me. --FvdP 00:32 Jan 25, 2003 (UTC)

 * More importantly, "brulée" looks wrong both to me and to my dictionary. --FvdP

 * Various other online dictionaries also seem to confirm "brûlée": can we have a cite for your paper dictionary, just to have a cite? The Anome

 * Dictionnaire Français-Anglais par Jean Mergault, Librairie Larousse, 1973. --FvdP 00:52 Jan 25, 2003 (UTC)

 * The Dictionnaire de l'Académie française, neuvième édition spells it crème brûlée.Mike Hayes (talk) 18:02, 14 June 2008 (UTC)

 * Stupid accented letters makes me glad of English's esoteric spelling and it's inconsistent pronunciation. At least I don't have to try to figure out how to get a grave accent on a letter. Mintguy

 * Simple: û = Alt+0251 (under Windows). ;-) --FvdP

 * yeah but what's that on my UK keyboard?

 * You don't have an Alt key ? Press it and keep it down. Press successively 0, 2, 5, then 1. Then release the Alt key. For the other letters, look at Start|Programs|Accessories|System_Tools|Character_Map. Select the wanted letter and search for "keystroke" on the bottom line of the window. I agree it's a tad annoying. --FvdP

 * There are many different keyboard encoding systems. The Unicode system is still not used universally, especially for non-Roman alphabets and the standard Unicode ALT + codes will not work for those systems.  You can change which system your keyboard uses in Control Panel if you want to. To find out what will work with your keyboard, go to, and type 'charmap', a box will open with all the characters in your encoding system, click on the character you want and it will tell you in the lower right corner what the ALT + keycode is Mike Hayes (talk) 18:02, 14 June 2008 (UTC).

 * Go to Google translate, and translate "burnt cream" into French and you get... crème brûlée The Anome

 * Come on guys! It is the dessert and its recipe that is important here, not its proper spelling.  Since the origins of the dessert are in contention we cannot settle on spelling until we settle on origin.  At our present state of knowledge lets go with common usage.  I vote with Mintguy and Anome.  The spelling "crème brûlée"  is also common in the US.  Too much unproductive energy is being expended on this debate! JSF 08:19 Jan 28, 2003 (UTC)

 * Oh yeah sure. Spelling in an encyclopedia doesn't matter. Mmmmm. Mike Hayes (talk) 20:12, 11 August 2008 (UTC)

 * There's a drop list under the edit box saying "insert". Choose "Latin". Click "û".  Easy.  It works on any keyboard and any operating system.  The Wednesday Island (talk) 20:51, 31 August 2008 (UTC)

Update:New spelling of Crème brulée

In 1990 the Académie française approved of new simplified spelling for French words and one of the changes was getting rid of circumflex accents on 'u' except in a few cases for disambiguation purposes. The official new spelling is Crème brulée. However, both old and new spellings are accepted in examinations under the rules of the French department of education. Which means that it can now be spelled either way. Both are correct. See w:fr:Annexe:Rectifications orthographiques françaises de 1990 Mike Hayes (talk) 20:58, 31 August 2008 (UTC)

 * Shouldn't we ask the folks on fr: to change the name of their article first? The Wednesday Island (talk) 21:05, 31 August 2008 (UTC)

 * The post above is not a suggestion that we change the name. It is information on official French spelling changes. It makes sense that articles in English on classic French food should use traditional spelling as that is the way it is spelled in the greater part of the literature and will continue to be for a long time. Getting rid of unnecessary accents make life simpler for those who write it every day but the traditional accents have greater aesthetic appeal on menus etc. and  I'm quite convinced that many francophones will continue to use them in that context. Mike Hayes (talk) 17:30, 1 September 2008 (UTC)

Turbinado sugar
Que'est-ce que c'est que "turbinado sugar"????? I'm baffled. Does it need a definition or is it just me being iggerant? :) Nevilley 21:37 Jan 18, 2003 (UTC)

 * A quick googling tells me it's US Equivalent of Demerara sugar. I substituted brown sugar. Mintguy

 * If you want to substitute, use regular granulated sugar. JSF

 * I just added a definition to the sugar article. -- Heron

 * Yum yum, well done and thanks. Deliberately, and contrary to normal practice, I've linked to sugar a second time in the article, to help people like me who might otherwise not have found that definition. ta! :) Nevilley 21:51 Jan 18, 2003 (UTC)

 * Heron's definition is indeed correct. Turbinado sugar is a blond, raw sugar generally available in major U.S. supermarkets.  I do not think that I would care for the stronger flavor of brown sugar in this delicately flavored dessert.

 * Terbinado is unheard of in the UK, it is virtually the same as Demerara sugar in the UK. Can I suggest - light brown sugar? Or - light brown sugar (Demerara or Turbinado)? Mintguy

 * What on earth is Demerara or Turbinado sugar ? (I make it with light brown sugar) Anthère

 * I agree, brown sugar sounds like a really bad idea. I've used turbinado and regular white.  Either way, the sugar needs to be easily granular in order to coat the surface prior to brulee'ing --24.5.197.145 (talk) 22:15, 1 July 2012 (UTC)

Fahrenheit
Do US recipe books still use fahrenheit? Everything is in Centigrade/Celsius in the UK and Europe. Mintguy

 * Yes, we still use Fahrenheit in the U.S. Dumb, aren't we!

 * The temperature window is very critical in this recipe. For the benefit of Centigrade users the window should read 77-79°C.

 * Yes we do, but in my opinion, the temperature of your oven isn't critically important if you're using a bain marie for your custard. As long as you're under like 450F (252C) and the ramekins are small, you can still get your custards to turn out nicely.  Off the top of my head, I'd say 350 would be a good target.  --24.5.197.145 (talk) 22:18, 1 July 2012 (UTC)

Measures

 * Hm. we need to convert these values to the international standard. -- Tarquin

I've found out that a US cup is 225ml liquid measure, but what is it in terms of grams for dry measure for sugar!, obviously it varies on whatever you're trying to measure? Also found out that a US tablespoon is 16ml, while a UK tablespoon is 15ml.
 * Mintguy
 * See also Cooking weights and measures

Mintguy, I will accept Turbinado or Demerara sugar, but not light brown, dark brown, or brown sugar. They are different sugar products, at least in the U.S. They have a much different flavor and texture than raw sugar. Also, brown sugars tend to lump, especially in a humid environment, so are difficult to spread evenly. I am assuming that Demerara sugar is coarse-grained raw sugar and not fine-grained brown sugar.

I have not mastered the editing protocol yet, so I have destroyed your sugar links in the recipe. You are free to reinstate them, but please don't change my raw sugar choices. Thanks for adding the Centigrade temperature window.

I will edit my recipe to present both the International and the U.S. measures. Thank you Tarquin for your inputs.

JSF 15:48 Jan 22, 2003 (UTC)

Trinity College story
After a bit of research prompted by Heron, and confirming the Delia claim in my own copy, I've found other references to it being invented at Trinity College in the late 18th century (at least 1769).

 * http://www.n1kp.com/main_menu.html
 * http://ae.boston.com/dining/recipes/t/trinity_cream.html
 * http://www.porters.uk.com/menu/default.asp?menuID=14
 * http://www.eng.umu.se/tt/proj1/recipes.htm
 * http://www.recipegoldmine.com/desspudd/desspudd49.html
 * http://www.hwatson.force9.co.uk/cookbook/recipes/desserts/cambridgeburntcream.htm

This site (http://emenus.ca/Victoria_and_Vancouver_Island/dictionary/C.htm) states the following Creme brulee ''It is a simple custard of nothing more than cream, eggs, sugar, and vanilla that is topped with a caramelized topping. History: The origins of this custard are very much in contention, with the English, Spanish, and French all staking claim. (1) The Spanish have taken credit for this dessert as crema catalana since the 18th century. (2) The English claim it originated in 17th century Britian in Trinity College, Cambridge. It is said that it was born when an English chef accidentally burned a custard he had sprinkled with sugar. The chef then passed it off as an original creation calling it burnt cream. It is also called trinity cream in England. Some old cookbooks even refer to it as Cambridge cream after Cambridge University.''

There seems to be little evidence that it's a Creole dish.

Also crème brûlée seems to be more common than other forms.
 * --Mintguy (no date given)

In a similar vein: The origins of crème brûlée are very much in contention, with the English, Spanish, and French all staking claim. The Spanish have taken credit for this sensuous custard as "crema catalana" since the eighteenth century, while the English claim it originated in seventeenth-century Britain, where it was known as "burnt cream" and the English school boys at Cambridge demanded it. It apparently wasn't until the end of the nineteenth century that common usage of the French translation came into vogue ... Debbie Puente in "Elegantly Easy Crème Brûlée" ( St. Martin’s Press). -- (unsigned, undated contribution)

 * Sheesh, Trinity folk aren't "school boys". The Wednesday Island (talk) 20:56, 31 August 2008 (UTC)

 * The only way to apply some scientific methodology to resolving this issue is to find historical recipes which can be dated, or other references in literature. Maybe someday when I have the time ... Mike Hayes (talk) 18:08, 1 September 2008 (UTC)

 * Most of the links in Mintguy's comment above are dead; the remaining ones are not WP:Reliable sources. I agree with Mike Hayes that we need to find reliable sources for all this. --Macrakis (talk) 18:40, 1 September 2008 (UTC)
 * I came across Creme de la creme by Bee Wilson, New Statesman, 21 May 2001, which refers to an essay by Elizabeth David. To look up.... --Macrakis (talk) 18:46, 1 September 2008 (UTC)

Clearing the kitchen

 * 1) Good Wikipedia food articles are about the history, sociology, meaning of food, in its context. Recipes are for the recipebook.
 * 2) Do be very skeptical of all recipe-origin tales that involve a "mistake" or an "unavailable ingredient." Some few are authentifiable: pommes de terre soufflés for example. --Wetman 07:14, 7 Dec 2004 (UTC)

Include recipes? (was: why I choose to revert)
What Wikipedia is not - ''Cookbook entries. For example, when writing an article about fried rice, don't give "A simple recipe for fried rice." That belongs in Wiki Cookbook (http://en.wikibooks.org/wiki/Cookbook). Instead, write an article about what is commonly included in a fried rice recipe, the history of fried rice, types of fried rice, how the Chinese and Japanese versions differ, etc.'' Gentgeen 18:17, 19 Jan 2005 (UTC)

 * Gentgeen, we had this discussion many many many times and you are absolute aware that there was no consensus to move all the recipes at all. There was on ongoing discussion about that, and there was NO agreement to do all the deletion you did on the recipe articles. You did it basically alone and boldly and people just gave up out of tiredness in front of your boldness; The ONE thing I would appreciate is that you do not try to pretend there is a policy on that topic, when there is no policy and you do not even have true community agreement to do so. I will doubly say that because I said on the french wikipedia that there is NO policy here on this topic, but only a bold person having a deletionist attitude and now I am accused of having lied. So, please, at least, recognise that pretending there is a policy is totally out of it. SweetLittleFluffyThing

In the category Wikipedia official policy, I find What Wikipedia is not. In that article, I find the text above. I did not add that policy to the category, I did not add that text to the policy, and I did not participate in the discussion about that text. It seems that I am not a lone deletionist acting boldly, but an editor simply following the officially established policy of this project. If you'll check the transwiki log, you will find that for a little over a year, several editors (including one of your fellow board members) have been involved in moving recipes to wikibooks, and listing recipes for deletion. To my knowldge, no one has left the project because I moved a recipe to wikibooks, but there is atleast one editor who has left the project because of the abuse you subjected her to because of this issue. Based on just your statement above, I'd have to conclude that you have misrepresented the issue. Gentgeen 19:50, 19 Jan 2005 (UTC)

 * who left the project because of me ? As for the rule, I removed it for now, since I could not find where it was discussed and agreed. If you wish to go on removing recipes fine, but it would be nice that it is not made a rule, as it is not supported by all the community. If a rule as reported right now, you would be right to revert or even to block me if I do not respect it. I think that would not be right to do so since there was no clear agreement all recipes should be deleted. I will add that I am pretty mad at this, because this is currently in discussion with a couple of guys on the french wikipedia, and I explained what happened here and that there was no decently agreed rule and many supporting them to stay here, and after your reverted I am accused of having lied. While I recognise your right to be bold, I do not accept well to be told I am lying when not aware of a rule which seems to come from no where. In short, I think that in "What wikipedia is not" there should be only rules that the biggest majority of people support. It is not the case, so ? Come back on irc to discuss this again please Gentgeen SweetLittleFluffyThing

 * I would like to see recipes for every dish as they serve as basis for defining the food especially when viewed from a vantage of actual pastry chef experience. For example, the line between a flan, a creme brulee and an egg custard are blurred without a recipe.  A recipe is a definition in bakery, look online for a recipe for a baguette for instance, the biggest difference in all the recipes is the adjustement for high alititude cooking.  In baking at least recipes should be re-added where they can be seen as having little deviation.  I am not a cook nor will I ever be because of the myraid ways in which dishes can be "cooked" and still called the same thing.  Looking over the chemistry pages I see many examples of preperation methods for things ranging from electrolysis to space age materials.  Having recipes that can genereally be agreed upon can help clarify where words cannot.  Excluding all recipes is like excluding all graphs, statistics or the like from any other page as they can not be agreed to either.  I presume Gentgeen is not in the food industry or he would see how lacking he has made these entries in some self-righteous ignorance that has got out of control.
 * --Rakista 15:57, 15 August 2005 (UTC)

Fluffy seems to have serious control issues. How many readers support your point of view? Have you taken a poll? An encyclopaedia contains descriptions of things, not instructions on how to produce them. That, in the simplest language possible, is the criterion. The problem cited by Rakista is already remedied by the link to Wikibooks. (I forgot to sign this post when I made it so I'm adding my signature now) Mike Hayes (talk) 18:03, 1 September 2008 (UTC)

Claim about liqueur
A 1-edit anon added a claim about caramelizing with liqueur. Though you can flambé anything (as one chef said, "the customers like it and it doesn't hurt the food too much"), I don't think that's standard.... --Macrakis 01:58, 16 December 2006 (UTC)

 * In the states, I've never heard of or seen this done and I've eaten very widely. It sounds unpredictable to me. --24.5.197.145 (talk) 22:22, 1 July 2012 (UTC)
 * Agreed. And not in French wiki either. Removed. Mcewan (talk) 22:32, 1 July 2012 (UTC)

Caramelization
The reference to caramelization (in addition to Maillard reaction) has been deleted. Is this accurate? Does this help the Wikipedia reader? If caramelization is not involved, surely this article should state why not. --Wetman (talk) 07:02, 23 June 2008 (UTC)

Merge from crema catalana?
Crème brûlée and crema catalana are essentially the same dish, and indeed there are claims (both ways) that one was the origin of the other. Wikipedia policy says that we organize articles by substantive topic, not by name, national associations, or point of view, so I see no reason for two articles here. Wikipedia policy further clarifies that we should prefer what the greatest number of English speakers would most easily recognize, which is quite clearly crème brûlée and not crema catalana, burnt cream, or Trinity cream. --macrakis (talk) 14:53, 12 January 2009 (UTC)
 * Strong agree Physchim62 (talk) 00:17, 8 February 2009 (UTC)

As a catalan who live near french I know well how is a creme brulee and crema catalana. Is well known the main ingredients are the same but the texture and the cook style is completly diferent, crema catala has a texture as custard but creme brulee is more like pie. Furthermore the flavour of the crema catalana is potenciated by cinamon unlike creme brulee which is powered by vainilla. RhinoFullmetal (talk) 20:48, 23 September 2020 (UTC)

Requested move

 * The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the move request was: Not moved. Jafeluv (talk) 13:52, 25 January 2010 (UTC)

Crème brûlée → Crème brulée — Also accepted in French, like crème fraiche.—Alwetendheid alom (talk) 16:06, 18 January 2010 (UTC)

 * Oppose Why? You haven't provided any reasoning. I prefer the version with the chapeau, if we're going to have diacritics, then use them. If not, then get rid of all of them. 76.66.197.17 (talk) 05:08, 19 January 2010 (UTC)
 * The point is that the circumflex on the u is not necessary, even in French; both crème brulée and crème brûlée are accepted. Why write this diacritic in English, when it does not have to be written even in French? Look at the German Wikipedia; it spells Crème brulée, which is the Duden spelling. We should always prefer the most simplified spelling in case of two equally accepted forms, as we do with crème fraiche. Alwetendheid alom (talk) 17:19, 19 January 2010 (UTC)
 * The current policy is to prefer the last stable state of the article in the event of two equally common forms, rather than to prefer simplicity.--Atemperman (talk) 03:37, 21 January 2010 (UTC)

 * Support diacritics seem right. —innotata (Talk • Contribs) 01:23, 20 January 2010 (UTC)

 * Oppose The lead makes it fairly clear that either is acceptable, and as a loanword we probably shouldn't be messing with it much. Some external sites seem to prefer the current diacritics usage anyway, which is good enough for me. — V = I * R (talk to Ohms law) 05:39, 20 January 2010 (UTC)

 * Oppose - what's changed since this was last discussed above? Knepflerle (talk) 04:13, 21 January 2010 (UTC)
 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

French IPA
What's the pronounciation?

 * kʁɛm bʁyˈle
 * kʁɛm bʁyle

70.29.211.138 (talk) 05:17, 21 January 2010 (UTC)

Since stress in French is not a property of words but rather of prosody (see here), French pronunciations should not have stress marks. --Atemperman (talk) 16:17, 22 January 2010 (UTC)

Removal of diacrictics
Since the user Alwetendheid alom has referred to the German Wikipedia as an example of the usage of Crème brûlée without certain diacritics: The German article was moved by him/her not too long ago. It has since been moved back the version with all diacritics in place. --Peter Putzer (talk) 22:11, 17 February 2010 (UTC)

Since the 1990 spelling reform, the correct spelling in French is without the circumflex over the u (Crème brulée). However, the traditional spelling is still acceptable. Since English has adopted the traditional spelling (with û), I support keeping the title unchanged. --ABehrens (talk) 17:21, 26 January 2020 (UTC)

Kerr story
This story appears in the current version:
 * Graham Kerr, in one of his 2001 Galloping Gourmet shows,[6] stated that a visiting Scotsman had given the Chef of Trinity College the recipe for "crème brûlée" in 1875, but that the Chef had 'kept it in his pocket' for four years, until finally introducing it in 1879.

It seems to me likely that Kerr (a Scot) was just being playful; he certainly isn't a food historian. I plan to remove this unless someone objects. --Macrakis (talk) 16:22, 7 August 2010 (UTC) Never mind -- I found a better source -- Elizabeth David quoting a near-contemporary report -- and included it in the article. --Macrakis (talk) 18:44, 7 August 2010 (UTC)

Origins
The page says the origins of this dish are unknown....What typical wikipedia twaddle. The desert was invented at Trinity College Cambridge in the 16th century, yes, Cambridge in England, not France. It came about when the students forgot to add sugar to their custard. they sprinkled sugar on the top and gently burnt it. To this day Trinity burnt cream is made with the topping stamped in the Trinity seal. Captainbeecher (talk) 18:23, 13 August 2010 (UTC)

 * Any reliable sources to back up this particular story? The furthest we've been able to trace the Trinity connection is to the 19th century. --Macrakis (talk) 16:16, 23 November 2010 (UTC)

Someone has reverted my changes on the origin of the dish. I wrote that blancmange is the predecessor of crème brûlée, based on a recent work of the French studious of medieval cuisine Eliana Thibaut Comalada. The person who reverted my change argued that by this rule of thumb, custard could then be considered the origin of the dish. Well, if it is the only argument to revert my contribution, I find rather weak, because custard is also preceded by blancmange. On the one hand, blancmange was reported as a dish in the first cuisine compendiums in the *Mediterranean area* in the XIII-XIV centuries and was soon very popular all over Europe due to translations and copies of these books.(For a reference, search for a book named "the book of Sent Sovi" of 1324) First records on custard come only 100 years later, in the XV century in England. At that time custard was more what we would now call an almond milk. Based on the fact that the first records of blancmange recommend to flavor it with almonds and that almonds came exclusively from the Mediterranean area at that time, it is not difficult to think that custard was a more liquid version of blancmange. I think this is a proof strong enough to say that blancmange is the predecessor of all these dishes.

And still now we can trace these "differences". blancmange used cornstarch, with high content of starch and so it is more thick. custard used almonds which have low content of starch and so it is more liquid. Nowadays, most people use cornstarch only and the game is in the proportions. If you boil 1 liter of milk with 40 grams of cornstarch and sugar, you have a blancmange. If you do the same with eggs and you burn sugar on top, you have creme brulee. If in the milk you also add cinnamon and lemon flavours, you have crema catalana. Now, if you prepare blancmange with less starch, you have a custard. If you add eggs in the custard, you have crème anglaise. If you add cinnamon and lemon flavours, hey! Noone has invented and recorded that. I will call that Unnilenni's creme :) Unnilenni (talk) 11:44, 23 November 2010 (UTC)

 * Clearly there is a large family of milk-based desserts. Blancmange is thickened with various starches; custard (in the modern sense) with eggs. Crème brûlée is a variety of custard with a hard caramel top.  The custard article already contains some information about the historical connection with blancmange-type preparations (thickened with starch) -- though it's a bit muddled.  Additional information about the blancmange-custard connection probably belong there.  On the other hand, there is no direct connection between blancmange and crème brûlée in particular as far as I can tell.  I don't see much point in repeating information which belongs in custard in the crème brûlée article. --Macrakis (talk) 16:16, 23 November 2010 (UTC)

US debut?
The Creme Brulee was first introduced by Sirio Maccioni in the US at the New York Restaurant called Le Cirque. —Preceding unsigned comment added by 71.183.61.196 (talk) 02:21, 27 March 2011 (UTC)

Technique
Is it true to say that one caramelises the sugar immediately before serving? If done under a grill rather than with a blowlamp it leaves the ramekins dangerously hot. I always allow about an hour between caramelising and serving. Also, ought we to discuss the method of making the custard? I always use a bain-marie in a low oven, but I have read (and have a citation for it) that the pukka way is in a saucepan, stirring diligently. I'd be frightened of curdling the stuff doing it that way.

Not for the article, but as a diversion for connoisseurs here, I have had crème brûlée served under the name of "spécialité du collège de Cambridge" in a restaurant in the rue des Petits-Champs in Paris. Tim riley (talk) 15:44, 4 March 2012 (UTC)

The sentence "Discs of caramel should not be prepared separately and put on top just before serving, or the caramel may be formed directly on top of the custard, immediately before serving." does not parse well for me. Should that first "not" actually be there? 196.210.19.93 (talk) 18:51, 8 October 2016 (UTC)

I changed the pic
So I changed the pic to a more aesthetic brulee. The exposure is worse, but I think the picture is better; both nicer to look at and also shows the custard with crust on the spoon. Vanilla bits are also visible in the custard, which is nice. If anyone disagrees, I'd be happy to discuss it. I'd also be really interested in seeing a slightly more square version of this photo with some cropping.

--24.5.197.145 (talk) 02:11, 1 July 2012 (UTC)

How different?
The article doesn't describe how crème brûlée is different from any other custard (other than the caramelized crust). Thick/thin/firm, smoothness, etc. Nor is it mentioned that it is heavy creme, not milk, that is used with the yolks to make crème brûlée. This all would probably be a rather helpful bits of information for a reader who has no idea what crème brûlée is. — al-Shimoni  (talk) 05:41, 12 April 2017 (UTC)

Recommended edits
Just some thoughts on the article from our group:

1. Reference 10: there's a lot of paraphrasing

2. Reference needed for: Analogous recipes appear in 17th-century Spanish cookery books, usually under the name of Cream of Saint Joseph, since it was a traditional dessert served during Saint Joseph's day.

3. Maybe add a section that talks about Modern-Day creme brulee ---> details about where to source ingredients/product (is it available pre-made in grocery stores?)

4. Some of the references aren't working

5. Article states that place of origin is Spain but in the details on the side it says France

Joakes02 AkhterMonir (talk) 18:56, 29 October 2017 (UTC) & aaliriani 18:51, 29 October 2017 (UTC)

Crema catalana
Most other wikis have separate articles for Creme brulee and Crema catalana. About the ref saying they're 'virtually identical', there are plenty of sources that disagree: To the dismay of French and Spanish, many persons think that Creme Brulee and Crema Catalana are the same things…--eh bien mon prince (talk) 15:57, 23 May 2018 (UTC)

 * What other wikis do isn't really relevant -- they are not reliable sources. Philosokitchen.com, similarly, is blog written by one non-specialist.
 * Multiple sources confirm that Crema Catalana is essentially the same dish as Crème Brûlée (though as with all foods, there are variations):
 * "virtually identical to the popular French dessert" (though as usual for this sort of recipe, people disagree on what is "authentic" or "original") Colman Andrews, Catalan Cuisine
 * "a similar recipe" Diane Toops, Eggs: A Global History
 * "The English, French, and Catalonians all lay claim to the origin of the dessert" ... "The Catalonian dessert crema catalana is made the same way" Oxford Companion to Sugar and Sweets
 * Best, --Macrakis (talk) 18:21, 23 May 2018 (UTC)

Merge from Crema catalana
As discussed before, Crema Catalana and Crème brûlée are virtually identical. To the extent that they do differ, that can of course be discussed in the article, as can the history. As for the title of the article, WP policy says that we use the most common English name. It may well be true that Crema Catalana is older, but that doesn't affect our naming policy. --Macrakis (talk) 18:45, 3 February 2020 (UTC)
 * This was recently split from creme brulee because of the fact it is misleading - it suggests that it's a different name for the same thing when, however similar they are, it clearly is not. As it is not the same dish, the policy on common name is irrelevant (not that it was relevant in the first place, as it doesn't apply to merging), because obviously the common name for this dish is its actual name. If there was an article on burnt cream dishes in Europe, maybe they could both be merged into that, but it's wrong to consider merging this dish into another one. Kingsif (talk) 21:29, 3 February 2020 (UTC)

 * Things do not need to be identical to be treated in the same article. For example, green sauce includes a variety of sauces which are based on chopped herbs; aioli include variants with and without eggs; etc.
 * In this case, the recipes are in fact almost identical. Both are sweet custards made of milk and eggs. Both have a crust of crisp caramel made by applying heat to sugar poured over the top. What's more, reliable sources tell us that they are the same (see above for quotations).
 * Apparently the only difference is the flavorings used -- the catalan version is flavored with lemon or orange and cinnamon, while the French version is flavored with vanilla.
 * Even the sources used in the current Crema catalana article indicate that they are almost identical. The lead sentence quotes a source saying that they are "virtually identical". Then the source for the different flavoring says "Although slightly different in texture and flavor, it's crème brûlée in spirit, down to its brûléed surface".
 * As for "If there was an article on burnt cream dishes"... this is that article. It is called crème brûlée simply because that is the common name for burnt cream dishes. --Macrakis (talk) 22:22, 3 February 2020 (UTC)

 * PS As a matter of procedure, please do not remove the merge template until there's been time for people to see it and discuss the topic. --Macrakis (talk) 22:23, 3 February 2020 (UTC)

 * See, I've eaten both, and they taste very different. Only people who don't know what the dishes are think they are the same. They're not, it's misleading to include them in the same article. Why you're on a crusade to do that, I don't know. PS Because your best argument for merge was something on common name that is irrelevant, the template can and should be removed. Kingsif (talk) 22:49, 3 February 2020 (UTC)

 * What they taste like to you is original research, and in any case, closely related foods and dishes can taste quite different, but still belong in the same article. Dark-roast coffee with cream and sugar tastes very different from light-roast coffee served black; my grandmother might spice meatballs with allspice and yours might spice them with red pepper, but they're still meatballs. Of course, the exact line is not completely clear -- we have separate articles for omelette, frittata, and eggah.
 * Since the lines aren't clear, we rely on reliable sources. And in fact we have a number of reliable sources saying that they are "slightly different" or "virtually identical" or "made the same way" and characterising them together as "the dessert", which is I think my "best argument". The link provided above by User:Underlying lk is an individual's blog, not a RS, and it claims that Crema catalana is made with milk and cornstarch, no eggs, so I think we can discount it.
 * Kindly assume good faith and don't attack me as being "on a crusade". Let's please focus on the sources and on applying WP's policies to this article. --Macrakis (talk) 03:19, 4 February 2020 (UTC)
 * I don't think this is significant enough to warrant discussion/response, but will note that since the discussion above on merging it was started by you two whole years ago, it seems like (if not a crusade) you at least have some shtick with this that nobody else does, for some reason. Kingsif (talk) 00:48, 5 February 2020 (UTC)

 * You are grossly misrepresenting the history. I merged Crema catalana into Crème brûlée in February 2009 after a merger proposal which received one Strong agree and no opposition after being open for over a month.
 * In May 2018, User:Underlying lk proposed splitting the articles, and pinged me. I responded to his arguments with WP:RS and that seemed to be the end of that. (This is what you are apparently referring to as a discussion on merging I started two years ago.)
 * Then, in October 2019, after ten years where the merger was stable, you unilaterally split the articles, moving content from one to the other with no discussion at all on Talk.
 * Given WP:BRD, I'd have been within my rights to revert your merger, but as a courtesy, I asked for a merger discussion. You responded to that discussion by claiming that it was "clearly" a different dish, providing no sources (reliable or otherwise), and removing the merger notice.
 * I again ask you to assume good faith, to avoid insulting me, and to follow WP policies. --Macrakis (talk) 17:12, 5 February 2020 (UTC)
 * PS, you might take a look at the Crème caramel article for comparison. It documents a number of variants (condensed milk, coconut, cream cheese, fruit, flavorings including vanilla, cinnamon, lemon peel, coffee, ...) with a number of names (flan, flan de leche, flan napolitano, purin, tocino de cielo, ...). --Macrakis (talk) 19:34, 6 February 2020 (UTC)

 * You had not responded to the above for 10 days, so it seemed that you no longer had objections. So I restored the Crema catalana material to this article. You reverted, with the comment "no consensus at tha talk page. Get consensus, you can’t propose a move, be told no, and still do it. Several times."
 * But in fact it was you who split the article in October with no discussion at all (let alone consensus), after 10 years where it was mostly stable. Not only that, when I put up a merger template pointing to the discussion, you deleted it, and proceeded to misrepresent my arguments, accuse me of being on a "crusade", and say I "have some shtick" with this article.
 * I provided WP:RS supporting a single article. I showed that the crème caramel article covers an even broader range of variants (as most recipes do). You provided only personal opinions and not a single RS. In the meantime, I have found some additional RS -- see Talk:Crème_brûlée/Crema_catalana.
 * At this point, it seems unlikely that further discussion will be productive, so I'll ask for a third opinion. --Macrakis (talk) 23:09, 16 February 2020 (UTC)
 * If there is no way to title the article (brûlée / catalana) because the 2 names are the defining aspect of the two...If there were a way to merge, how would the article be titled, since the 2 identifying names differ by the fact of being 2 different cultures, the two cultures didn't provide a way to think the two things are the same, because nationality (language (of France) and specific place (Spain)... https://en.wikipedia.org/w/index.php?title=Crema_catalana&diff=271014002&oldid=270442633 cannot be acceptable because the name of the dessert belongs to, is a possession of Spanish culture Diametakomisi (talk) 23:41, 16 February 2020 (UTC)

Saw this at WP:3O, but it seems like if a merger proposal is properly formatted, those templates should do the work of alerting people to discussion. It looks like mergefrom was missing, however, so I added it. I've looked at the sources cited in the other article, and did my own search. The overwhelming majority of the sources about the subject (as opposed to sources which are just recipes) make a connection to creme brulee. As such my inclination is that they are similar enough (or, more specifically, covered by sources as similar enough) to justify a merge. I have a lingering concern, though: is this sort of coverage typical across languages? Certainly creme brulee is very common here in the US, and perhaps in other English-speaking countries such that it makes sense that something with similar properties would be covered with it. But I wonder if that's the case internationally... — Rhododendrites  talk \\ 23:42, 16 February 2020 (UTC)
 * additionally, for example, in Draft:List_of_discoveries_in_mathematics "(Harriot 1601,[5] Snell 1619) Brahmagupta", both the discoverer and the subsequent discoverers are mentioned Diametakomisi (talk) 23:45, 16 February 2020 (UTC)

The mergefrom and mergefrom have been up now for many months, with no additional discussion. Re the point about the name, WP policy is clear (WP:OTHERNAMES). Articles use one title, and alternatives are mentioned in the text of the article. The content of Crema catalana was incorporated into Crème brûlée a long time ago. So I will change Crema catalana to a redirect to it. --Macrakis (talk) 19:07, 28 June 2020 (UTC)

 * I've only just now come across this merger, as a result of looking up crema catalana. Knowing both crema catalana and crème brûlée well, and thus aware that the two are, from ingredients through preparation to the method for achieving burnt effect... to taste, noticeably distinct, it would not have occurred to me that one would have been buried in the article labeled for the other. They are not alternative names; they are not the case of Gdańsk and Danzig, not alternative spellings, longer or shorter forms, historical names, etc.
 * Some of the discussion above jumps the rails a bit, but the overall effect is that no consensus is reached for folding crema catalana into crème brûlée. The two could be sub-types of a descriptively factual over-arching category, but that's not what's done here (analogy: manchego and pecorino are "cheeses made from sheep's milk", but one is definitely not a type of the other). One of the positive aspects of Wikipedia is that contents are constantly being refined in terms of accuracy. The merger of these two is a move in the opposite direction, and runs the risk of implicitly misinforming -- and certainly misleading -- ingenuous readers, a counter-current to the purpose of an encyclopedia. Barefoot through the chollas (talk) 15:48, 28 August 2020 (UTC)
 * As the discussion above and the relevant Wikipedia policies say, two items do not need to be identical or to be two names for the same thing to be discussed in the same article. I hope you'll agree that crème brûlée, crema catalana, and Trinity cream have a strong family resemblance. There is a custard base (with varying flavorings), and there is a hard caramel top. This is in effect the "descriptively factual over-arching category" you mention above -- it just happens that there is no standard name for it: it would not be helpful (or consistent with Wikipedia policy) to rename this article, say, Custard desserts topped with hard caramel, because that is not a commonly-used name. I'll note, by the way, that we do have a nice source -- from a Catalan cookbook -- that crema catalana is "virtually identical" to crème brûlée. Even if it is not, it makes more sense to have a single article.
 * Almost all dishes have a range of variations, and many of them are considered to be national or regional variants. We should celebrate the variation in the article; but that doesn't need multiple articles. --Macrakis (talk) 17:27, 28 August 2020 (UTC)
 * I'm going to, perhaps, disappoint by confessing that I'm not interested in a pointlessly endless bickering match. No harm in observing the obvious, though: an article such as Custard desserts, sort of parallel to the Green sauce model, would work fine. Summary descriptions of typologies (and subtypes, if useful) with appropriate links to the individual articles. If real consensus finds that Custard desserts or something similar is not common or clear enough (although I'd never heard of green sauce as such in English), well, c'est la vie, no need to stretch for a higher node. If an intermediate burnt custards or similar is ruled out, Crema catalana and crème brûlée stand well as separate articles, with cross-linking, and bundling them together loses more than it gains, for reasons explained clearly above. This really isn't a difficult case. Barefoot through the chollas (talk) 19:07, 28 August 2020 (UTC)
 * Discussion is how we resolve differences on WP. Calling it "pointless endless bickering" hardly moves the discussion forward.
 * The article custard is mostly about custard desserts, though it also mentions some non-dessert custards. Besides plain custard, it also includes crème caramel and crème brûlée as well as some other things. I don't see a need for a separate custard dessert article; do you?
 * I wouldn't mind renaming this article burnt custard, but our policy says to use the most common name, and crème brûlée is about 200x more common than burnt custard. For that matter, burnt custard seems to be primarily UK usage, and on the decline (and some of the hits in ngrams are talking about custard that has been unintentionally burnt, not custard with caramel on top).
 * Green sauce is an interesting comparison. The issue there is that salsa verde in the US at least has a well-established meaning as a particular Mexican sauce which is (amusingly, strangely) not a member of the category green sauce, and in any case is only 2x as common, not 200x.
 * Re "this isn't a difficult case", nice to know that you agree with yourself, but the goal of a discussion is to convince others. --Macrakis (talk) 19:41, 28 August 2020 (UTC)
 * "the goal of a discussion is to convince others." In a word... no. And therein the rub. As long as the goal is viewed as being to convince others rather than to come to agreement on a satisfactory solution, stalemate will be the result. As for the rest, sorry, not falling into this trap. Barefoot through the chollas (talk) 20:56, 28 August 2020 (UTC)
 * You're right. But asserting "this isn't a difficult case" hardly advances toward agreement. --Macrakis (talk) 21:16, 28 August 2020 (UTC)

 * Oppose There are benefits to keeping separate articles instead of one article "burnt custard". This source even says that crema catalana does not have to be topped with caramel. . Spudlace (talk) 21:12, 28 August 2020 (UTC)
 * Oppose Agreed. In addition to facts and argumentation above, I've just noticed that the German-language Wikipedia handles these nicely. Two separate articles, each referring to the other. Looks as though more than one person agrees with himself that das ist nicht schwer. Barefoot through the chollas (talk) 21:25, 28 August 2020 (UTC)

Point of order There is no open merge discussion. As you'll see from the history above, there was one article covering both variants of hard caramel custard since 2009, with a brief interlude where an editor split them without discussion. I asked if there were any objections to restoring the status quo ante in February, and only got one objection, from the person who unilaterally split it without discussion. So I restored the status quo ante in June. If you want to start a "split" proposal now, go ahead, and add an appropriate announcement to the page. (Obviously I would be opposed.) --Macrakis (talk) 22:24, 28 August 2020 (UTC)

Now back to substance. I have had both dishes, too, and certainly there are differences, but there is much more in common than there is different. Grilled lamb chops (paidakia) as served in Greece (say) are very different from grilled lamb chops as served in a New York steakhouse -- they are cut differently, seasoned differently, grilled differently, and garnished differently -- but they're recognizably in the same category. So I'd say that there are two issues: should one article cover both variants? And if so, what should be its title? In the spirit of "coming to an agreement", I would much rather have one article titled "Burnt cream custard" or some such than have two articles, even if it technically violates WP:NAMINGCRITERIA. In my opinion, splitting up families of similar foods along national lines is destructive of the value of WP. The next logical step would be to separate out Trinity cream, which has its own recipe and its own mythology. --Macrakis (talk) 22:24, 28 August 2020 (UTC)

 * There's just no policy reason to merge French, Spanish, and English cuisine articles. These European cuisines have well known and published primary source traditions that have become firmly established in Western academic literature, and often have distinct cultural traditions and associated customs that have developed over centuries. Spudlace (talk) 22:42, 28 August 2020 (UTC)
 * Right. And the model exists. These work nicely:, . Barefoot through the chollas (talk) 23:27, 28 August 2020 (UTC)
 * On the contrary. If a family of dishes is found in more than one cuisine, it should ideally be treated in a single article. It would be absurd to have a separate article for French pommes frites, Belgian pommes frites/frieten, British chips, American French fries, etc., even though they have different names and differ in a variety of ways -- the typical fat used, the typical size, the most common condiments used with them, the most common dishes they accompany, etc. The policy reasons are WP:NOTDICT (don't separate by name, but by topic) and WP:CFORK (don't separate by point of view).
 * Often, of course, there is significant variation among varieties, sometimes even along national lines. But reifying "national cuisines" by splitting articles about closely related families of foods is not helpful. Should we also have an article about Spanish Crema de San José? What exactly would it say that wasn't redundant?
 * So the question is whether crème brûlée and crema catalana are different enough to require different articles. I think we all agree that they are custard desserts topped with hard caramel, right? We also agree that the typical flavorings used in the caramel differ. But surely that is just as true of, say, rice pudding. Why would we have separate articles for Greek rice pudding (typically flavored with cinnamon), Levantine rice pudding (with rosewater), etc.?
 * Moreover, we have multiple reliable, third-party sources saying that the two dishes are "virtually identical".
 * At the same time, there are many things we can do to improve the coverage of the Catalan version in this article. For example, the article says that crema catalana is documented in the 14th-century Llibre de Sent Soví, but without a precise page reference which would let us verify it. It would surely be worthwhile to have a blockquote or feature box with the full 14th-century recipe. --Macrakis (talk) 15:44, 29 August 2020 (UTC)
 * Café au lait, Café con leche, Caffè latte, Wiener Melange, Cappuccino, Galão, Caffè macchiato, Cortado, Latte macchiato... Barefoot through the chollas (talk) 17:25, 29 August 2020 (UTC)
 * Excellent example of a set of articles that should be merged! Two wrongs don't make a right. --Macrakis (talk) 18:34, 29 August 2020 (UTC)
 * I undid the merge which was done without achieving a consensus. The merge should not be done while the discussion is open. Spudlace (talk) 20:27, 29 August 2020 (UTC)

 * Huh? The stable version is actually the merged version. It was split with zero discussion in 2019 after almost 10 years in that form. I put a merge notice up in February to restore that version and it remained up until June, when I restored the stable version. Now, suddenly, two months after that, it is apparently so urgent to split the article that you've cut off discussion and restored the undiscussed split version after just one day's discussion? --Macrakis (talk) 20:47, 29 August 2020 (UTC)
 * Spudlace, thanks for straightening this out. Text in both can now be cleaned up a bit. Barefoot through the chollas (talk) 21:31, 29 August 2020 (UTC)

 * Oppose Living myself in Northern Catalonia (in France), all restaurants here mostly have Crema catalana. Those that do more traditional French cooking do Crême brûlée. The difference between the two is an evidence for all customers and no one would call Crême brûlée Crema catalana or the opposite. Culex (talk) 23:22, 26 September 2020 (UTC)

"Creme broolay" listed at Redirects for discussion
A discussion is taking place to address the redirect Creme broolay. The discussion will occur at Redirects for discussion/Log/2021 November 3 until a consensus is reached, and readers of this page are welcome to contribute to the discussion. Un assiolo (talk) 19:46, 3 November 2021 (UTC) I am late to the party but I personally would support it: it fits the "regular" 26 letters of English alphabet. OkiPrinterUser (talk) 15:06, 5 June 2023 (UTC)

No Creme Brulee in French cookbooks until the 1980s (?)
Is there really no creme brulee in the basic works of, say, Raymond Oliver (since ca. 1958), Bocuse or Lenotre? - I haven't got those anymore, so can't look myself.--Ralfdetlef (talk) 06:57, 10 December 2022 (UTC)

 * Absence of proof is not a proof of absence. Please try the correct French spelling. OkiPrinterUser (talk) 15:09, 5 June 2023 (UTC)

title: Talk:Crème brûlée


text:
Talk:Symmetric matrix

Shouldn't be better to create a distinct entry for 'skew-symmetric matrix' ?

Inverse Matrix
Does the inverse of a square symmetrical matrix have any special properties? Does being symmetrical provide any shortcut to finding an inverse? 58.107.136.85 (talk) 03:56, 11 April 2008 (UTC)

If the inverse of a symmetrical matrix is also a symmetrical matrix it should be stated under properties. —Preceding unsigned comment added by 77.13.24.86 (talk) 18:25, 25 January 2011 (UTC)

Yes! Of course the inverse of a symmetric matrix is symmetric; its very easy to show too.

Proof:

Suppose A = A^t and A is non-singular, then there exists A^-1 such that A*A^-1 = I. Applying the transposition operator to each side of the equation we get...

Transpose[A*A^-1] = Transpose[I]... {(A^-1)^t}*A^t = I; however, we have that A = A^t, so it follows that... {(A^-1)^t}*A = I, but the inverse is unique therefore,... (A^-1)^t = A^-1. This proves that the inverse is symmetric. QED — Preceding unsigned comment added by Brydustin (talk • contribs) 00:36, 1 January 2012 (UTC)

Basis, Eigenvectors
It's easy to identify a symmetric matrix when it's written in terms of an orthogonal basis, but what about when it's not? Is a real-valued matrix symmetrix iff its eigenvectors are orthogonal? —Ben FrantzDale 00:31, 11 September 2006 (UTC)
 * Reading more carefully answers my question: "Every symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix." So apparently the answer is yes. —Ben FrantzDale 15:27, 11 September 2006 (UTC)

I believe you're confusing a couple of concepts here. A matrix is a rectangular array of numbers, and it's symmetric if it's, well, symmetric. Of course, a linear map can be represented as a matrix when a choice of basis has been fixed. On the other hand, the concept of symmetry for a linear operator is basis independent. Greg Woodhouse 01:34, 30 November 2006 (UTC)

 * being symmetric with real entries implies unitarily diagonalizable; the converse need not be true. anti-symmetric matrices with real entries are normal therefore unitarily diagonalizable. but the eigenvalues are no longer real, so one must speak of unitary matrices, rather than orthogonal. Mct mht 04:07, 12 September 2006 (UTC)

It's been a while since I followed up on this. I still feel like there is something missing in this article. For me back in 2006, I was confused about the importance of symmetry of a matrix because they are "just" rectangular arrays of numbers. As such, symmetry seems like a superficial property that can be undone by simple things like swapping rows. Furthermore, we could have a matrix that is symmetric but meaninglessly so. For example, a data matrix of participants with age and weight as columns. If Alice is 80 and weighs 90 pounds and bob is 90 and weighs 80 pounds, then you get a symmetric table, but that symmetry doesn't mean anything (for starters, the units don't match, but we could construct something for which they did). That left me wondering "when does symmetry mean something?" I now think I understand. Consider the moment matrix of a bunch of points in R3. That is a symmetric 3×3 matrix. As I've come to understand things, that matrix is contravariant (in the tensor sense) in its rows and columns.

I think matched variance of rows and columns is a necessary (but not sufficient) condition for a matrix to be symmetric in any meaningful sense. That implies that a meaningfully symmetric matrix is strictly-speaking the matrix representation of a tensor. Does that sound right? (I don't mean to say that [80 90; 90 80] isn't symmetric, I am just saying that for that symmetry to be anything other than coincidence, the matrix has to have matched variance in rows and columns.) —Ben FrantzDale (talk) 13:46, 14 December 2010 (UTC)

"More precisely, a matrix is symmetric if and only if it has an orthonormal basis of eigenvectors" This statement is just wrong. See 'Normal Matrix'. Normal matrices need not be symmetric (in fact they can be anti-symmetric), but does have an orthonormal basis of eigenvectors. However, it IS true that if a matrix is symmetric, then it has an orthonormal basis (in fact this is trivially true, since all 'symmetric matrices' are 'normal matrices', and normal matrices have an orthonormal basis of eigenvectors) Please correct. —Preceding unsigned comment added by 128.122.20.210 (talk) 03:17, 30 December 2010 (UTC)

It is correct if we assume that eigenvectors are real. Then A=O^TDO, and A^T=O^TD^TO=A. This is a bad username (talk) 22:16, 8 February 2016 (UTC)

Symmetric matrices are usually considered to be real valued
I've made several changes to indicate that symmetric matrices are generally assumed to be real valued. With this, the real spectral theorem can be stated properly. VectorPosse 05:03, 12 September 2006 (UTC)
 * Would it be better to have a little more detailed discussion of Hermitian? --TedPavlic 16:21, 19 February 2007 (UTC)

It may be worthwhile to add a section on complex symmetric matrices, or matrices that are (complex) symmetric w/r/t an orthonormal basis. They are not as useful as self-adjoint operators, but the category includes toeplitz matrices, hankel matrices and any normal matrix. 140.247.23.104 04:43, 12 January 2007 (UTC)

 * I agree. We just need to make sure it's in a different section so that it doesn't get mixed up with the stuff about the spectral theorem.  VectorPosse 19:28, 19 February 2007 (UTC)

Products of Symmetric Matrices: Eigenspaces Closed Under Transformation
As the article states, products of symmetric matrices are symmetric if and only if the matrices commute. However, it also says, "Two real symmetric matrices commute if and only if they have the same eigenspaces." This makes no sense. Consider arbitrary matrix $$A$$ and the identity matrix $$I$$. Certainly, $$A I = I A$$, so these matrices commute. However, in general $$A$$ and $$I$$ will not have the same eigenspaces! I think this statement was supposed to be, "Two real symmetric matrices commute if and only if they are simultaneously diagonalizable," or, "Two real symmetric matrices commute if and only if the eigenspace for one matrix is closed under the other matrix." Both of these statements sound complicated compared to the original statement. I'm not sure if it's worthwhile to even mention it. However, I'm going to make a change. I'm okay with someone removing the statement entirely. --TedPavlic 17:34, 19 February 2007 (UTC)

 * the previous version was correct. two real symmetric matrix commute iff they can be simultaneously diagoanlized iff they have the same eigenspaces. please undo your change. Mct mht 10:24, 21 February 2007 (UTC)

 * As far as I can see, Ted's counterexample (identity matrix and arbitrary symmetric matrix) shows that two symmetric matrices can commute without having the same eigenspaces. Please tell me where we go wrong. -- Jitse Niesen (talk) 11:25, 21 February 2007 (UTC)

 * hm, that depends on what's meant by "having the same eigenspaces", no? if that means "the collection of eigenspaces coincide", then you would be right. (however, seems to me the wording of the comment, which i removed, about the "closure" of eigenspaces can be improved.) perhaps it's more precise to say two real symmetric matrix commute iff there exists a basis consisting of common eigenvectors. Mct mht 12:17, 21 February 2007 (UTC)

 * also, the identity matrix is really a degenerate case. since it and its multiples are the only matrices that's diagonal irrespective of the basis chosen. excluding such cases (if A restricted to a subspace V is a · I, remove V), seems to me that the general claim is true: real symmetric matrices {Ai} commute pairwise iff the family of eigenspaces of Ai and the family of eigenspaces of Aj are the same for all i and j. Mct mht 15:42, 21 February 2007 (UTC)

 * I agree with "two real symmetric matrices commute iff there exists a basis consisting of common eigenvectors". I think the more common formulation is "two real symmetric matrices commute iff they are simultaneously diagonalizable", so I'd prefer that. I agree that the formulation "the eigenspace for one matrix is closed under the other matrix" is rather unfortunate as I had to read that sentence a couple of times before I understood what is meant.
 * I don't understand what you mean with "if A restricted to a subspace V is a · I, remove V". Every matrix is a multiple of the identity when restricted to an eigenspace, and after removing the eigenspaces of a symmetric matrix there's nothing left. -- Jitse Niesen (talk) 04:04, 22 February 2007 (UTC)

 * shoot, you're right. well, remove V if dimension V is > 1. that better? Mct mht 04:10, 22 February 2007 (UTC)

 * hm, forget it Jitse, that did not make it better. you're right there. Mct mht 12:20, 22 February 2007 (UTC)

Hey..the definition of symmetrizable matrices is not complete. A symmetrizable matrix is a product of a symmetric matrix and a positive definite matrix. The positive definite matrix need not be a invertible diagonal matrix as in the section. Please check... Naik.a.s —Preceding unsigned comment added by Naik.a.s (talk • contribs) 10:08, 27 July 2009 (UTC)

eigenvalues
are the eigenvalues of A:n×n,  A=AT  always  {0,...,0,tr(A)} ? applies for matrix BTB with B=[1,2,3,4] --Saippuakauppias ⇄ 10:48, 31 December 2007 (UTC)

 * No. For instance, the identity matrix is symmetric, but has eigenvalues {1,1,…,1}. However, every matrix of the form $$A=B^TB$$ does have {0,…,0,tr(A)) as its eigenvalues. Such matrices are called rank-one matrices, because their rank is one. -- Jitse Niesen (talk) 15:24, 31 December 2007 (UTC)

In the article the statement "Two real symmetric matrices commute if and only if they have the same eigenspaces." is wrong. For a counterexample consider the identity matrix and any diagonal matrix with more than one eigenvalue. The statement should read: "If two real symmetric matrices of dimension n commute then a basis for R^n can be chosen so that every element of the basis is an eigenvector for both matrices."

Incidentally the answer above is is assuming that B is itself a rank one matrix (as in the example given with B=[1,2,3,4]). It's not true for B an arbitrary matrix.

137.222.137.107 (talk) 15:16, 15 June 2012 (UTC)Nick Gill

The spectral theorem...
...is conspicuous by the absence of any mention of it in this article!

Maybe I'll be back. Michael Hardy (talk) 02:18, 10 August 2008 (UTC)

 * It's at the start of the "Properties" section. -- Jitse Niesen (talk) 10:57, 10 August 2008 (UTC)

trace of the product of three matrices
Hi,

there's a mistake in the article. It's claimed that the trace of three symmetric (or hermitian) matrices is invariant under arbitrary permutations. To prove this, it's used that (CBA)^t = CBA which is simply not true because the product of symmetric (hermitian) matrices is symmetric (hermitian) if and only if they commute. —Preceding unsigned comment added by 192.33.103.47 (talk) 09:27, 28 June 2010 (UTC)

Complex symmetric matrix eigenvalues
Hello, the page currently states that for each complex symmetric matrix, there exists a unitary transformation such that the resulting diagonal matrix has real entries. The eigenvalues of complex symmetric matrices are generally themselves complex, and not all real. Somebody tell me if I'm reading this wrong, otherwise I'm going to change the wording "is a real diagonal matrix" to "is a complex diagonal matrix". — Preceding unsigned comment added by Jeparie (talk • contribs) 18:36, 25 September 2015 (UTC)

 * The entries of the diagonal matrix are the singular values not the eigenvalues of the original matrix. Best wishes, --Quartl (talk) 19:47, 25 September 2015 (UTC)

The article is still (or again) wrong. A complex symmetric matrix doesn't necessarily have real eigenvalues, as the article currently states in the Decomposition section. Either we need to change complex symmetric matrix to complex Hermitian matrix, or elaborate that the diagonal matrix doesn't contain eigenvalues. — Preceding unsigned comment added by 213.52.196.70 (talk) 18:07, 6 November 2017 (UTC)

Math notation
I changed all the math from inline math to tex. This was reverted by 87.254.93.231 several times. The article as it is now contains an ugly mixture of tex and inlne math. I propse changing all again to tex. Reasons for Tex: Reasons for inline math
 * You can display almost everything with Tex
 * Tex is easy to distinguish from surrounding nom-math text. E.g. for a matrix $$A$$ vs. for a matrix A.
 * Less work to write article

What do you think? 11:36, 15 January 2019 (UTC)

Proposing adding new proof of Takagi
''Note: This proof yields a short algorithm for computing the Takagi decomposition using software packages like Scipy and Matlab. The schur decomposition feature should be used for this.''

Let $$\operatorname{uncomplexify} : M_n(\mathbb C) \to M_{2n}(\mathbb R)$$ denote the injective ring homomorphism $$A + Bi \mapsto \begin{pmatrix}A & -B \\ B & A \end{pmatrix}$$, where we use block matrix notation. Let $$M = A + Bi$$ be an arbitrary non-singular complex-symmetric matrix. (The non-singularity restriction will later be lifted.) Observe that while the matrix $$\operatorname{uncomplexify}(i M) = \begin{pmatrix}-B & -A \\ A & -B \end{pmatrix}$$ is not $$\mathbb R$$-symmetric, we may define $$J = \begin{pmatrix}0 & I \\ I & 0 \end{pmatrix}$$ and get that $$\operatorname{uncomplexify}(i M) J = \begin{pmatrix}-A & -B\\ -B & A \end{pmatrix}$$ is $$\mathbb R$$-symmetric. By the spectral theorem it follows that $$\operatorname{uncomplexify}(i M) J$$ has an orthonormal eigenbasis $$v_1, v_2, \dotsc, v_{2n}$$ with real eigenvalues $$\lambda_1 \geq \lambda_2 \geq \dotsc \geq \lambda_{2n}$$. Let $$K = \begin{pmatrix}0 & -I \\ I & 0 \end{pmatrix} $$ and observe that for every eigenvector $$v_i $$ in our eigenbasis, the vector $$K v_i $$ is an eigenvector of eigenvalue $$-\lambda_i $$. We therefore improve our orthonormal eigenbasis of $$\operatorname{uncomplexify}(i M) J$$ to the new orthonormal eigenbasis $$(v_1,v_2,\dotsc,v_n,Kv_1,\dotsc,Kv_2,Kv_n) $$, which we interpret as a block matrix $$P$$. Verify that each $$v_i$$ is indeed orthogonal to each $$K v_j$$ because they have distinct eigenvalues (which might fail if $$M$$ were singular). Given our definition of $$P$$, we have that $$\operatorname{uncomplexify}(i M) J = P \begin{pmatrix} \Lambda & 0 \\ 0 & -\Lambda\end{pmatrix} P^T$$ (where $$\Lambda = \operatorname{diag}(\lambda_1,\lambda_2,\dotsc,\lambda_n)$$). Observe that $$P $$ is equal to $$\operatorname{uncomplexify}(U) $$ for some matrix $$U $$; this follows because if we write each $$v_i$$ as the block matrix $$\begin{pmatrix}w_i \\ \hline z_i\end{pmatrix}$$, we have that $$P = \begin{pmatrix}W & -Z\\ \hline Z & W \end{pmatrix} = \operatorname{uncomplexify}(W + iZ)$$. Observe also that $$U$$ is unitary; this follows because $$\operatorname{uncomplexify}(I) = I = P^T P = \operatorname{uncomplexify}(U)^T \operatorname{uncomplexify}(U) = \operatorname{uncomplexify}(U^* U)$$, and by cancelling $$\operatorname{uncomplexify}$$ due to injectivity, we get $$U^* U = I$$. We therefore have that

$$\begin{aligned} \operatorname{uncomplexify}(iM) J &= P \begin{pmatrix} \Lambda & 0 \\ 0 & -\Lambda\end{pmatrix} P^T \\&= \operatorname{uncomplexify}(U) \begin{pmatrix} \Lambda & 0 \\ 0 & -\Lambda\end{pmatrix} \operatorname{uncomplexify}(U)^T \\&= \operatorname{uncomplexify}(U) \operatorname{uncomplexify}(i\Lambda) J \operatorname{uncomplexify}(U)^T \\&= \operatorname{uncomplexify}(U) \operatorname{uncomplexify}(i\Lambda) \operatorname{uncomplexify}(\overline U)^T J \\&= \operatorname{uncomplexify}(U i \Lambda \overline U^*) J \end{aligned}$$

Cancelling $$J$$ (as it's invertible) and then cancelling $$\operatorname{uncomplexify}$$ (because it's injective) yields $$M = U \Lambda U^T$$.

The result can be extended to any singular matrix $$M$$ by approximating $$M$$ as a sequence of invertible matrices $$M_n$$, and then forming the pair of sequences $$(U_n, \Lambda_n)$$ such that $$M_n = U_n \Lambda_n U_n^T$$. Since the components of $$U_n$$ and $$\Lambda_n$$ (for each $$n$$) are bounded, we may appeal to the Bolzano–Weierstrass theorem to get a pair of subsequences $$(U_{n_k}, \Lambda_{n_k})$$ that both converge. We therefore get $$M = \lim_{k \to \infty} M_{n_k} = \left(\lim_{k \to \infty}U_{n_k}\right)\left(\lim_{k \to \infty} \Lambda_{n_k}\right)\left(\lim_{k \to \infty} U_{n_k}\right)^T$$. We are done. --Svennik (talk) 14:54, 19 May 2022 (UTC)

Proof 3 of Takagi
Note: This proof does not treat singular matrices as a special case.

Let $$\operatorname{uncomplexify} : M_n(\mathbb C) \to M_{2n}(\mathbb R)$$ denote the injective ring homomorphism $$A + Bi \mapsto \begin{pmatrix}A & -B \\ B & A \end{pmatrix}$$, where we use block matrix notation. Let $$M = A + Bi$$ be an arbitrary complex-symmetric matrix. Observe that while the matrix $$\operatorname{uncomplexify}(i M) = \begin{pmatrix}-B & -A \\ A & -B \end{pmatrix}$$ is not $$\mathbb R$$-symmetric, we may define $$J = \begin{pmatrix}0 & I \\ I & 0 \end{pmatrix}$$ and get that $$\operatorname{uncomplexify}(i M) J = \begin{pmatrix}-A & -B\\ -B & A \end{pmatrix}$$ is indeed $$\mathbb R$$-symmetric. Let $$K = \operatorname{uncomplexify}(iI) = \begin{pmatrix}0 & -I \\ I & 0 \end{pmatrix} $$ and observe that for every eigenvector $$v$$ of eigenvalue $$\lambda$$ of $$\operatorname{uncomplexify}(i M) J$$, the vector $$K v $$ is an eigenvector of eigenvalue $$-\lambda$$. We therefore build an orthonormal eigenbasis of $$\operatorname{uncomplexify}(i M) J$$ in the following way:

 * We start with the empty basis $$B$$ and the linear map $$L : \operatorname{span}(B)^\bot \to \operatorname{span}(B)^\bot, v \mapsto \operatorname{uncomplexify}(i M) J v$$.
 * While $$\operatorname{span}(B)^\bot$$ is not $$0$$-dimensional, we let $$v$$ be a unit eigenvector of $$L$$. Observe that as well as $$v$$ being orthogonal to $$\operatorname{span}(B)$$, so is $$Kv$$. We replace $$B$$ with $$B \cup \{v,Kv\}$$, and replace $$L$$ with its restriction to $$\operatorname{span}(B)^\bot$$.

We arrange our resulting eigenbasis into the block matrix $$P = (v_1,v_2,\dotsc,v_n,Kv_1,Kv_2,\dotsc,Kv_n)$$. Given our definition of $$P$$, we have that $$\operatorname{uncomplexify}(i M) J = P \begin{pmatrix} \Lambda & 0 \\ 0 & -\Lambda\end{pmatrix} P^T$$ (where $$\Lambda$$ consists of the eigenvalues corresponding to the eigenvectors $$v_1$$ to $$v_n$$). Observe that $$P$$ is equal to $$\operatorname{uncomplexify}(U) $$ for some matrix $$U $$; this follows because if we write each $$v_i$$ as the block matrix $$\begin{pmatrix}w_i \\ \hline z_i\end{pmatrix}$$, we have that $$P = \begin{pmatrix}W & -Z\\ \hline Z & W \end{pmatrix} = \operatorname{uncomplexify}(W + iZ)$$, so that $$U = W + iZ$$. Observe also that $$U$$ is unitary; this follows because $$\operatorname{uncomplexify}(I) = I = P^T P = \operatorname{uncomplexify}(U)^T \operatorname{uncomplexify}(U) = \operatorname{uncomplexify}(U^* U)$$, and by cancelling $$\operatorname{uncomplexify}$$ (due to injectivity) we get $$I = U^* U$$. We therefore have that

$$\begin{aligned} \operatorname{uncomplexify}(iM) J &= P \begin{pmatrix} \Lambda & 0 \\ 0 & -\Lambda\end{pmatrix} P^T \\&= \operatorname{uncomplexify}(U) \begin{pmatrix} \Lambda & 0 \\ 0 & -\Lambda\end{pmatrix} \operatorname{uncomplexify}(U)^T \\&= \operatorname{uncomplexify}(U) \operatorname{uncomplexify}(i\Lambda) J \operatorname{uncomplexify}(U)^T \\&= \operatorname{uncomplexify}(U) \operatorname{uncomplexify}(i\Lambda) \operatorname{uncomplexify}(\overline U)^T J \\&= \operatorname{uncomplexify}(U i \Lambda \overline U^*) J \end{aligned}$$

Cancelling $$J$$ (as it's invertible) and then cancelling $$\operatorname{uncomplexify}$$ (because it's injective) yields $$M = U \Lambda U^T$$. --Svennik (talk) 19:46, 19 May 2022 (UTC)

Symmetric Hessian
In the section on the Hessian it might be worth mentioning that there is a quite crucial assumption made there which is the twice differentiability. As the Hessian of a function can be well defined even if it is not twice differentiable (see for example https://calculus.subwiki.org/wiki/Failure_of_Clairaut's_theorem_where_both_mixed_partials_are_defined_but_not_equal#Example) and then the partial derivatives do not commute and the Hessian is not symmetric. It is stated there that it applies to twice differentiable functions but it might be worth explicitly writing that it does not apply to all functions whose Hessian is well defined. Tom-Lukas Lübbeke (talk) 14:52, 6 July 2024 (UTC)

title: Talk:Symmetric matrix