Talk:Logarithm of a matrix

But the numerical method is only really possible for real symmetric matrices, no? Charles Matthews 22:00, 18 October 2005 (UTC)

I think it is okay as long as the matrix is diagonalizable (though I seem to remember that it is unstable). I also removed your statement on 2x2 matrices which I didn't quite understand by the statement that every nonsingular matrix has a matrix logarithm. Hopefully, I'll manage to write a bit more later. -- Jitse Niesen (talk) 23:35, 18 October 2005 (UTC)

PS: I'm glad to see you throw your hat in the ring again. -- Jitse Niesen (talk) 23:40, 18 October 2005 (UTC)

It occured to me that Charles might have been thinking about matrix logarithms that are real, while for me the logarithm can well be complex. -- Jitse Niesen (talk) 10:41, 19 October 2005 (UTC)

So, for real symmetric positive-definite there is no issue with taking logs of the eigenvalues. Anything else: well, it might give

log λ

with λ < 0, which is 'interesting'. Otherwise you can of course have complex eigenvalues, or not be able to diagonalize. Probably with a real matrix and a pair of conjugate complex eigenvalues something good happens.

The 2×2 case

${\begin{bmatrix}a&b\\-b&a\end{bmatrix}}.$

ought to be the same question as the complex logarithm. Charles Matthews 10:46, 19 October 2005 (UTC)

More explanation: I believe exp is surjective from n×n complex matrices to invertible matrices; but it is not surjective from n×n real matrices to invertible real matrices, which is not even connected. So in discussing what kind of inverse function there is, you do really need the complex entries. Also, it will only be a local inverse function. Charles Matthews 10:55, 19 October 2005 (UTC)

Yes, that's correct. I'll think a bit about the 2x2 case. I remember from when I studied this stuff, that some things do not quite work out as I'd expected. For example, the matrix

${\begin{bmatrix}-1&0\\0&-1\end{bmatrix}}$

does have a real logarithm, namely

${\begin{bmatrix}0&\pi \\-\pi &0\end{bmatrix}}.$

Geometrically, this corresponds with rotation through 180 degrees.

By the way, what do you think about the statement that the matrix logarithm "is in some sense an inverse function of the matrix exponential"? I'm not very happy with it because it is imprecise (in what sense?), but I think some sentiment like this needs to be expressed in the lead section.

Finally, on rereading I realized that my PS above could be rather mysterious. It refers to your standing for the ArbCom election. -- Jitse Niesen (talk) 12:32, 19 October 2005 (UTC)

The diagonalization

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)

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)

The article emphasizes the algorithmical point of view. It would be nice to strengthen the connection to matrix Lie groups and Lie algebras. Geometric examples help to understand why the logarithm is usually multi-valued. I included one such example.

--193.175.8.13 (talk) 16:53, 1 February 2008 (UTC)