Talk:Logarithm of a matrix

message: As noticed before me, this calculation of the matrix logarithm works only for diagonalizable matrices. That has to be menioned in the article, no? Conceptually though, any invertible matrix should have a logarithm, that follows from functional calculus, but I guess one can't find the log so easily for nondiagonalizable matrices. Oleg Alexandrov (talk) 13:28, 19 October 2005 (UTC)
replies: text: Yes, we can actually boost the article and make it more interesting by getting those extra points of view in. On nilpotent matrices, exp is a polynomial mapping to unipotent matrices, with polynomial inverse; in some sense, then, the off-diagonal block of the Jordan form is simpler, and numerically. If you already have the Jordan form of M, then you have it as a product diagonal×unipotent, with commuting factors, so log takes multiplication to addition. Of course a numerical analyst doesn't want to do that first; but it is clarifying, I think. Charles Matthews 15:07, 19 October 2005 (UTC)
thread_title:  The diagonalization 

message: The article emphasizes the algorithmical point of view.
It would be nice to strengthen the connection to matrix Lie groups and Lie algebras.
I started a section on this.
Geometric examples help to understand why the logarithm is usually multi-valued.
I included one such example.
--Benjamin.friedrich (talk) 22:06, 2 February 2008 (UTC)
thread_title:  Connection to Lie groups 

message: Currently (2009-02-28) the page *seems* to have ab contradiction (2) "Properties" claims that the M.Log. exists whenever the matrix is invertible, whereas (10) "Constraints in the n x n case" says that *additionally* all Jordan blocks corresponding to negative Eigenvalues must appear even times.
If you look closely, the latter is about *real* matrices, but nowhere is it mentioned that the former is about general complex matrices. Is it?
Maechler (talk) 15:15, 28 February 2009 (UTC)
thread_title: Real-complex question

message: The edits by R.e.b. have been undone because they detract from the article. Discussion has been going on about the domain of logarithm when applied to real matrices. See 2 × 2 real matrices#Functions of 2 × 2 matrices for a larger context. It is important to be correct. Further, it is helpful to the reader to indicate that logarithm of matrices gets one into the door of Lie theory. Why R.e.b. said this is an error is a mystery.Rgdboer (talk) 03:23, 4 November 2009 (UTC)
thread_title:  Real matrices 

message: Almost every time this article mentions a logarithm of a matrix M, it refers erroneously to the logarithm of M.  This is just incorrect, since every matrix that has a complex logarithm also has more than one complex logarithm. This is misleading to the reader, who is led to believe incorrectly that matrices in general have a particular matrix that is their logarithm. This is not true in any natural sense. Except for special categories of matrices, such as positive definite symmetric matrices M (whose "principal" logarithm log(M) can be taken as the diagonal matrix with entries that are the real logarithms of the eigenvalues of M).
This ought to be fixed in the article.2600:1700:E1C0:F340:9C6D:A392:5AC0:7D43 (talk) 04:08, 28 June 2018 (UTC)
replies: text: The first half of the article (up to and including the fourth section on Existence) diligently points out, again and again, that logarithms are not unique. But after that the article becomes less consistent in its assumptions (real or complex), notation, etc. So I agree that the later sections could use some work. Mgnbar (talk) 19:05, 28 June 2018 (UTC)
thread_title:  Problem with article 

message: IP editor 77.102.101.85 has twice now edited the start of the "Power series expression" section, in my opinion making the text worse, both in its prose and in its correctness. But, before this turns into a revert war, maybe other editors would like to chime in. Mgnbar (talk) 22:28, 19 November 2018 (UTC)
thread_title:  2018 November 19 

message: It is not clear to me which matrix norm this article refers to when it states that the power series expression converges if ||B-I||\<1. The source Hall (2015) uses ||B|| to denote the Hilbert-Schmidt norm trace(B*B)^(1/2), so perhaps that is the right one. But could one use another matrix norm instead? In any case, I would find it helpful if the article would specify the permissible matrix norms for the statement on the convergence of the power series.
Aliceschwarze (talk) 01:18, 6 March 2019 (UTC)
replies: text: You are right that the article should clarify the norm. I don't have the required reliable sources in front of me. Meanwhile, I do suspect that it's the norm you mentioned, which in the real case is called the Frobenius norm (or the Euclidean norm). Mgnbar (talk) 13:01, 7 March 2019 (UTC)

text: Any sub-multiplicative matrix norm (such as any matrix norm induced from a vector norm) will do. The proof is simple, since then \|B^k\|\le\|B\|^k for any k. Equivalently, it is sufficient that the spectral radius is less than 1. I'll add this soon. McKay (talk) 23:27, 7 March 2019 (UTC)
replies: text: Oh nice. I think this should be mentioned explicitly (perhaps in parentheses), since I had the same concern as Aliceschwarze upon reading. Ei283 (talk) 01:30, 10 January 2024 (UTC)
thread_title:  Issue with section on power series expression: matrix norm 

message: Specifically, if \left\|{B-I}\right\|<1, then the preceding series converges and e^{\mathrm{log}(B)}=B.
In what norm the condition \left\|{B-I}\right\|<1 is?
132.64.39.238 (talk) 11:23, 8 March 2020 (UTC)
replies: text: For the answer, see the text immediately above your question. (Though these details should be added to the article itself by someone with reliable sources at hand.) Mgnbar (talk) 17:54, 8 March 2020 (UTC)
thread_title:  What norm? 

message: The integral doesn't make sense. Perhaps it's meant for each "1" to be "I". I don't know what's meant. Somebody who does should fix it. Getthebasin (talk) 19:53, 28 January 2021 (UTC)
replies: text: I agree. It's not clear why there are scalars in the numerators and matrices in the denominators. Perhaps "1 / BLAH" simply means BLAH^-1. In any event, it should be clearer. Mgnbar (talk) 22:03, 28 January 2021 (UTC)

text: In addition, I would love to see a citation for this integral, so I can learn more about where it comes from Mrfout (talk) 00:38, 21 October 2021 (UTC)
replies: text:  https://www.ias.edu/sites/default/files/sns/files/1-matrixlog_tex(1).pdf Cuzkatzimhut (talk) 11:39, 21 October 2021 (UTC)
replies: text: So the equation in Properties seems to be equation (6) in that PDF. Thanks. Still, it would be nice to have a reliable source. (This source does not appear to be peer-reviewed, explicitly states that it is not rigorous, and even cites Wikipedia.) Mgnbar (talk) 13:41, 21 October 2021 (UTC)
replies: text:  Whatever; these notes are more than reliable, if only you understood what a towering giant the IAS author is. The notes are more rigorous than the level of the article. Fussing about rigor while stumbling on the obvious is bad form. The parting "reference" on WP's  Matrix exponential#The exponential map simply reminds you of the well-known formula. WP provides a ref to Willcox's article. This is routine stuff. Cuzkatzimhut (talk) 15:14, 21 October 2021 (UTC)

text:  Appreciate the addition, thanks! Mrfout (talk) 12:45, 31 December 2021 (UTC)
thread_title:  Integral formula error in the Properties section 

message: I'm not sure about this, but if the summation goes from k=1  to infinity, shouldn't the first element be (-1)^{2} \frac{(B-I)^2}{2}, which is the second one in th formula? The formula includes the one for k=0 so I think either the lower bound of k or the formula is wrong. DoerteMitHut (talk) 12:39, 16 June 2021 (UTC)
replies: text: You're talking about the power series expansion? It looks right (or at least self-consistent) to me. Notice that, although the exponent on -1 is k + 1, the denominator and the exponent on (B - I) are simply k. Mgnbar (talk) 14:54, 16 June 2021 (UTC)
thread_title:  first element of the sum