Talk:Normal matrix

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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)

[edit] 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)

[edit] 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)

[edit] '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)