Talk:Normal matrix

message: Should "Hermitian" be written with a capital 'H'? I think it always is everywhere else I've seen it, but I don't know what accepted style is for this. --DudeGalea 16:22, 19 Jun 2005 (UTC)
thread_title: Capitalization

message: Correction  : a positive definite matrix is not normal ! Take a small 2 by 2 example to see that :
$$A=\left(\begin{array}{cc}5&1\\0&7\end{array})\right.$$ The matrix $A$ is positive definite but non normal.
:The matrix \begin{bmatrix}1&1\\0&1\end{bmatrix} is an even simpler example.  I think the original author assumed positive definiteness implied symmetry.  This isn't necessarily the case for real matrices.  (See the article on positive definite matrix.)  Lunch 04:17, 13 July 2006 (UTC)
thread_title: Positive definiteness

message: The "analogy" section of this page states that it is "occasionally useful" to make the analogy with complex numbers - but it doesn't give any explanation of why or in what way it can be useful.  Without such information the statement of the analogy doesn't seem very helpful.
Nathaniel Virgo (talk) 17:13, 10 October 2009 (UTC)
replies: text: I didn't add that section, but after reading it I found it useful. If nothing else, it helps understand what the defintions mean. :) Shreevatsa (talk) 20:01, 10 October 2009 (UTC)
thread_title:  Analogy section 

message: In Joel Franklin's classic 1968 Matrix Theory, he defines normal matrices as the matrices which have orthogonal eigenvectors, and then proves that these matrices are characterized by the property that they commute with their adjoint.  This article does the reverse: it defines normal matrices as those matrices that commute with their adjoint, and then as a "consequence" show that these matrices have orthogonal eigenvectors.  Neither approach is "right" or "wrong", but it may be worth checking other literature to see if there is a consensus. Lavaka (talk) 19:12, 15 April 2010 (UTC)
thread_title:  chicken vs egg 

message: How widely is normality used as a test for diagonalizability? I personally found the second paragraph confusing since it seems to imply that many or most diagonalizable matrices are normal, and normality is a quick and easy way to test for it.
"Normality is a convenient test for diagonalizability: every normal matrix can be converted to a diagonal matrix by a unitary transform, and every matrix which can be made diagonal by a unitary transform is also normal, but finding the desired transform requires much more work than simply testing to see whether the matrix is normal."  --Aluchko (talk) 20:49, 14 May 2012 (UTC)
replies: text: Normality is equivalent to unitary diagonalizability, which is stronger than diagonalizability. But to actually find the unitary transform that diagonalizes the matrix means to find the eigenvectors, which from what I hear can be numerically tricky. Mct mht (talk) 03:30, 15 May 2012 (UTC)
thread_title:  Test for diagonalizability 

message: "The entire space is spanned by some orthonormal set of eigenvectors of A."
I could not find any material stating this, but rather statements equivalent with the eigenvectors of A spanning the entire space. If so, the vectors are generating the space, thus a basis can be selected. If a basis can be selected, then via the Gram-Schmidt orthogonalization, an orthogonal set of vectors can be constructed, which are still the eigenvectors of the matrix, and normalizing them does not change that. Is this reasoning sound? If so, wouldn't it be better to elaborate a bit in the article?
Andorxor (talk) 01:58, 5 January 2013 (UTC)
replies: text: Orthonormality follows trivially from that unitary diagonalizability of a normal matrix. Having a set of eigenvectors which forms a basis (mere diagonalizability) is strictly weaker than normality. Mct mht (talk) 02:08, 5 January 2013 (UTC)

text: The article already mentions that the columns of U are eigenvectors. They are orthonormal because U is unitary.  Conversely, given an orthonormal basis of eigenvectors, let U be the matrix having the basis vectors as columns; it is unitary, by orthonormality. In the product AU every column is scaled by its eigenvalue, i.e., AU = UΛ. Multiply both sides by U*. Certainly a reader new to the subject might not see this immediately, but to me this point does not stand out as needing a citation among other statements of the same nature in the article. Tag removed.  Styrofoams (talk) 17:05, 23 March 2014 (UTC)
thread_title:  Orthonormal set of eigenvectors 

message: I have undone changes, which introduced the following text:
There is an obvious misunderstanding of the terms "special case" and "sufficient". Since I am not an expert in this field, I decided to remove the contribution instead of rectifying it.
209.141.188.140 (talk) 01:20, 20 July 2019 (UTC)
replies: text: Thanks for spotting the error and sorry for the trouble. I got mixed up when I was writing the first paragraph... Is the set of normal matrices that are a subset of EP matrices. But looking back now and seeing the EP matrices article, I think this article is better without this contribution. Saung Tadashi (talk) 20:39, 20 February 2021 (UTC)
thread_title:  EPness and normality 

message: In the classification, Dirac and Pauli matrices should be in the same category, since they are so similar.--Chris2crawford (talk) 04:55, 7 February 2022 (UTC)
thread_title:  Matrix Classes panel at the end 

message: I have doubts about the last sentence of comment in source in #Special cases
:but among 2×2 matrices, there are only ones that are multiples of unitary matrices.
The matrix A=\begin{bmatrix}1&-1\\1&1+i\end{bmatrix} is a normal matrix, but its eigenvalues have unequal modulus 1.90211, 1.17557, so A is not a multiple of unitary matrix. Hbghlyj (talk) 23:06, 28 November 2022 (UTC)
thread_title:  2×2 normal matrices are multiples of unitary matrices 

message: I suggest getting rid of the fancy boxes.  They contribute nothing to the article except a graphic design, which is unnecessary.  (I find them intrusive.)  Opinions?  Zaslav (talk) 10:36, 3 December 2023 (UTC)
replies: text: 👍 I haven't checked out the boxes on this page, but on other pages I've seen colored boxes cause the equations to be illegible when viewed in dark mode on mobile. The-erinaceous-one (talk) 07:00, 4 December 2023 (UTC)
thread_title:  Fancy boxes