Talk:Diagonalizable matrix

message: 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)
thread_title:  when is P to be normed ? 

message: 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)
replies: text: They can be diagonalized, yes, but they can't be simultaneously diagonalized unless they commute. Algebraist 19:31, 22 January 2010 (UTC)

text: 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)

text: 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)

text: 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)
thread_title:  Simultaneous diagonalization  

message: 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)
thread_title:  Incorrect characterization 

message: 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)
thread_title:  Other 

message: 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)
thread_title:  Move to Diagonalization 

message: 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)
thread_title:  Rotation matrix