Talk:Diagonalizable matrix

Moving to Diagonalizability ?
What would people think of moving this article to "diagonalizability"? 193.48.101.101 20:18 15 Jun 2003 (UTC)


 * That is a better name for it. The artlicle should be named after the property. Padde 22:03, 20 June 2006 (UTC)


 * Why? I think "diagonalizable matrix" is clearer and more specific. The word "diagonalizability" is not used a lot. -- Jitse Niesen (talk) 03:49, 21 June 2006 (UTC)


 * Furthermore, there are many other articles with names like square matrix, invertible matrix, positive definite matrix, normal matrix, etc. Diagonalizable matrix follows this convention. Eclecticos 04:20, 21 September 2006 (UTC)

I find that the necessary conditions for a diagonalizable matrix could be defined in a way that is comprehensible for people withouth a math degree, no? Anton, 02.11.2007
 * Everything should be as simple as possible, but no simpler. - Albert Einstein.
 * Furthermore, I'm an undergrad CS student and I find the conditions expressed in a way such as they are perfectly comprehensible by everyone that should/could care what "diagonalizable" means. akbg (talk) —Preceding comment was added at 23:51, 15 May 2008 (UTC)

when is P to be normed ?
Is it right, that P is never to be normed, but it's gonna be easier to calculate P-1, since then, if P=orthogonally => $$P^{-1}=P^T$$ ? On my calculator (TI-89), however, he norms P always, when searching the EigenVectors ... --Saippuakauppias ⇄ 09:13, 16 January 2008 (UTC)

Simultaneous diagonalization
the article says simultaneous diagonalization to be equivalent to commutation. This cannot be strictly right, but I do not have my copy of Horn&Johnson here to check, do that in a few days. Counterexample: Two symmetric (or hermitian) positive-definite matrices can always be diagonalized, commutation is not necessary! --Kjetil Halvorsen 19:27, 22 January 2010 (UTC)
 * They can be diagonalized, yes, but they can't be simultaneously diagonalized unless they commute. Algebraist 19:31, 22 January 2010 (UTC)

Answer to algebraist: They _can_ be simultaneously diagonalized, but not by an orthogonal (or unitary) matrix! See the section I just added to the article on Positive-definite matrix. --Kjetil Halvorsen 22:07, 22 January 2010 (UTC) —Preceding unsigned comment added by Kjetil1001 (talk • contribs)
 * Suppose A and B can be simultaneously diagonalized, say as A=PDP-1 and B=PEP-1, with D and E diagonal. Then AB=PDP-1PEP-1=PDEP-1=PEDP-1 (since diagonal matrices evidently commute) = BA. Algebraist 22:12, 22 January 2010 (UTC)
 * Ah, I see you're using transpose rather than inverse in the definition of diagonalizability. This is not standard, and I can't immediately see why it's not downright wrong. Do you have a source for it? Algebraist 22:14, 22 January 2010 (UTC)

Too Algebraist: What I have done is standard, and very useful, theory (one application is optimization of a quadratic form under a quadratic form constraint where it is useful too diagonalize the criterion matrix and the constraint matrix simultaneously). One reference for the theory is the very same Horn&Johnson, 1985, but now pages 218 & following, section 4.5:" Congruence and simultaneous diagonalization of Hermitian and symmetric matrices." You should read that one! very specifically, look to the last paragraph on page 227, starting with: " There are several types of simultaneous diagonalization results that we might consider. We might have two Hermitian matrices A and B and we might wish to have UAU^* and UBU^* diagonal for some unitary matrix U, or ... ". The possibilities are summarized in theorem 4.5.15.

I give you one day to comment on this, and then I will go on to revert your reversal, and expand on the material, including this Horn&Johnson reference. --Kjetil Halvorsen 21:57, 27 January 2010 (UTC) —Preceding unsigned comment added by Kjetil1001 (talk • contribs)
 * It would be nice if this article had a section on other notions of diagonalization, such as diagonalization of quadratic forms. All we seem to have at present is some stuff at quadratic form and Sylvester's law of inertia. Perhaps things like the Smith normal form should be mentioned as well. But the bulk of the article should stay devoted to the primary topic, the diagonalization of (matrices representing) linear self-maps of a finite-dimensional vector space, and anything referring to another meaning should be clearly labelled as such. Algebraist 16:25, 30 January 2010 (UTC)

I do agree that anything referring to another meaning should be referred as such, although in the literature this advice it not always followed! But making it clear that there are multiple concepts of diagonalization will confuse readers less. --Kjetil Halvorsen 17:13, 30 January 2010 (UTC)

Incorrect characterization
The recent edits to the Characterization section where incorrect. They said...
 * An n-by-n matrix A over the field F is diagonalizable if and only if the sum of the dimensions of its eigenspaces is equal to n, which is the case if and only if there exists an orthonormal basis (or indeed any basis) of Fn that consists of eigenvectors of A. If such an orthonormal basis has been found, one can form the matrix P having these basis vectors as columns, and P−1AP will be a diagonal matrix. The diagonal entries of this matrix are the eigenvalues of A.

In particular, the basis does not have to be orthonormal, and it's easy to construct an example where that condition can't be enforced. For instance, use a 2x2 matrix that sends (1, 0) to (1, 0) and (1, 1) to (2, 2). If A were Hermitian, then the basis could be orthonormal, as is discussed at the end of the Diagonalization section. 24.220.188.43 (talk) 00:01, 11 August 2011 (UTC)

Other
I don't know what "resp." means. Is it standard terminology? I suggest it should be defined. — Preceding unsigned comment added by Blitzer99 (talk • contribs) 20:26, 27 September 2011 (UTC)

Is it just me, or is the quantum mechanical application at the end of the article a little bit far-fetched/out-of-place? — Preceding unsigned comment added by 128.214.137.193 (talk) 14:49, 15 August 2013 (UTC)

Move to Diagonalization
The Diagonalization has many meanings and the disamibguation page brings here, into diagonalizable matrix page, when I click the Matrix diagonalization. So, the matrix diagonalization is a part of diagonalizability topic here. Yet, I think it is vice-verse. People need diagonalization and work out the diagonalizability, as part of it. --Javalenok (talk) 13:21, 2 January 2013 (UTC)

Rotation matrix
The section "Matrices that are not diagonalizable" currently begins with a statement linking rotation matrix twice. Recent edits have had various changes to this particular statement. Since I have already contributed to this tussle, this venue must serve. Suggestion to raise the field when there are no real eigenvalues seems inappropriate. If you can, please attend.Rgdboer (talk) 01:55, 22 February 2013 (UTC)