Talk:Normal matrix

Too many terms in this stub are not defined. This article either needs expanding to explain the terms, to have the terms converted into links where they are explained or the be merged into the main matrix page.

My matrix stuff was too many years ago for me to be able to fix this. -- SGBailey 22:19 Jan 17, 2003 (UTC)

Capitalization

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)

Positive definiteness

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)

'Hermitian' vs 'hermitian' etc.

(1) Such words are sometimes uncapitalized if they are being used intensively, by specialists in a particular area. But I don't think 'Hermitian' is usually in this category. In an encyclopaedia, I would leave the capital in. (Likewise with 'Laplacian', 'Lagrangian' etc.)

(2) Positive definite. Horn and Johnson,'Matrix Analysis', p396, make being Hermitian (and therefore symmetric, in the case of real matrices) part of the definition of being positive definite. A misjudgment, surely? - it certainly confused me for a while. Note to self or request to others - check Wikipedia's article on positive definiteness.

(3) Some more simple examples of matrices that are not normal would aid understanding. NTSORTO - insert some discussion accordingly.

Rwb001 7 March 2007

(1) I agree.

(2) I had some discussion about this on Talk:Cholesky decomposition. This led to the impression that the definition of Horn and Johnson is common, but not universal. Therefore, it's best to be explicit on Wikipedia, instead of relying on the definition being used in positive-definite matrix. Anyway, the latter article does not say clearly whether Hermitian is required, precisely because the literature is confused about this issue.

(3) I tend to agree, but I'm not totally sure what you mean and I don't fully grasp the concept of "normal matrix".

Jitse Niesen (talk) 09:58, 7 March 2007 (UTC)

(2) for complex matrices, <Ax, x> ≥ 0 for all x implies that A is Hermitian, but this is not true in general for real matrices. perhaps Horn and Johnson required Hermiticity in both the real and complex cases to get a uniform definition?

(3) the unilateral shift T is an example of a non-normal operator. the commutator T*T - TT* is a rank-1 projection. for a similar finite dimensional example, consider the square matrix that is 1 on the superdiagonal and 0 elsewhere. Mct mht 10:37, 7 March 2007 (UTC)

Analogy section

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)

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)

I think the point is that a normal matrix is similar to a diagonal matrix with complex entries. If all of the entries are real/imag/unit/positive then the matrix itself is Hermitian/AnitHermitian/Unitary/Positive. Since addition and multiplication of diagonal matrices act individually on each entry, normal matrices share the same properties as the corresponding complex numbers. This only works for more multiple matrices if they are simultaneously diagonalizable, however.Chris2crawford (talk) 04:50, 7 February 2022 (UTC)

Perhaps something like above could be added to this section. There are already a few similar Propositions in the "Consequences" section, but they are separately stated and not tied together, which obscures the generality of the normal matrix analogy.Chris2crawford (talk) 04:50, 7 February 2022 (UTC)

chicken vs egg

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)

Equivalences

There seems to be a problem with equivalence number 10: "A commutes with some normal matrix N with distinct eigenvalues" if and only if A is normal. I think this is only true if A has N distinct eigenvalues. If A is normal with some degeneracy, any matrix commuting with A will share its eigenspaces, and therefore share its degeneracy. Can anyone check this for me please? Bobathon71 (talk) 22:32, 28 July 2011 (UTC)

Distinct eigenvalues means N has one dimensional eigenspaces corresponding to each eigenvalue. So the statement is OK. The fact that two normal matrices commute say nothing about their kernels. Take, for example, the identity matrix and the zero matrix. Mct mht (talk) 12:19, 30 July 2011 (UTC)

Ok... I think I'm with you. Just checked the proof I was looking at. If A commutes with B and B is normal, then every k-dimensional eigenspace of A has to contain an orthogonal set of k eigenvectors for B. I interpreted that as eigenspace of B, which it obviously doesn't have to be. B could have N distinct eigenvalues and still satisfy that condition. Thanks. (Let me know if I'm not getting it) Bobathon71 (talk) 18:42, 30 July 2011 (UTC)

Let's look at it this way. Take a normal matrix B. If B has, say, 3 eigenvalues λ₁, λ₂, and λ₃ with multiplicities k₁, k₂, and k₃ respectively. So in some orthonormal basis, B takes the form

$B={\begin{pmatrix}\lambda _{1}I_{k_{1}}&0&0\\0&\lambda _{2}I_{k_{2}}&0\\0&0&\lambda _{3}I_{k_{3}}\end{pmatrix}}$

where $I_{k}$ denotes the k by k identity matrix. Now, in the same basis, a matrix A commutes with B if and only if A is of the form

$A={\begin{pmatrix}M_{k_{1}}&0&0\\0&M_{k_{2}}&0\\0&0&M_{k_{3}}\end{pmatrix}}$

where $M_{k}$ is a k by k matrix. If B has distinct eigenvalues, then each k_i is 1, i.e. A is unitarily diagonalizable, hence normal. Mct mht (talk) 05:07, 31 July 2011 (UTC)

Thanks. That's a good way of putting it. A is then diagonalisable in the unique basis that diagonalises B... but not necessarily uniquely. If A has any multiple eigenvalues then the diagonal form of A will contain blocks of $\lambda I_{k}$, and any change of basis among the degenerate eigenvectors will also leave A diagonal. Bobathon71 (talk) 10:15, 31 July 2011 (UTC)

Eigenvectors of commuting normal matrices

Can anybody link a proof (or references) of the statement "the columns of U* are eigenvectors of both A and B"? (last phrase of section "consequences")— Preceding unsigned comment added by 77.168.20.183 (talk) 13:56, 22 December 2011 (UTC)

If $\mathbf {A}$ is diagonizable by a unitary matrix $\mathbf {U}$, then $\mathbf {UAU^{*}} =\mathbf {\Lambda }$, with non-zero elements ${\Lambda }_{ii}=\lambda _{i}$.

Pre-multiplying by $\mathbf {U^{*}}$, and using its unitary property, we have $\sum _{j=1}^{n}A_{ij}(U^{*})_{jk}=\sum _{j=1}^{n}(U^{*})_{ij}{\Lambda }_{jk}=(U^{*})_{ik}{\lambda }_{k}$.

This is an eigenvalue equation $\mathbf {Au_{k}} =\lambda _{k}\mathbf {u_{k}}$, where $\mathbf {u_{k}}$ is the $k$th column of the matrix $\mathbf {U^{*}}$

You can do the same with $\mathbf {B}$.

Bobathon71 (talk) 01:44, 6 January 2012 (UTC)

Test for diagonalizability

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)

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)

Orthonormal set of eigenvectors

"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)

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)

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)

EPness and normality

I have undone changes, which introduced the following text:

EP matrices are also a special case of normal matrices. EPness, however, is not a sufficient condition to normality. For example,

$A={\begin{pmatrix}1&2&0\\3&4&0\\0&0&0\end{pmatrix}}$

is an EP matrix, but not normal.

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)

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)

Matrix Classes panel at the end

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)

2×2 normal matrices are multiples of unitary matrices

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)

Fancy boxes

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)

👍 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)