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Let $a, b \in \mathbb{R}$. We know that $\log(ab) = \log(a) + \log(b)$. My question is very straightforward:

Is there an equivalent function for one of the two most used matrix products? I.e., does there exist a function $f: M_n(\mathbb{R}) \to M_k(\mathbb{R})$ for some $k$ such that either:

  1. $f(MN) = f(M) + f(N)$; or
  2. $f(M \otimes N) = f(M) + f(N)$, where $\otimes$ denotes the Kronecker product;

I've never heard of any such function, though I believe the matrix exponential does the job when going from Kronecker sums to products, and the same for commuting matrices in the regular product. Perhaps this means at least one of the sums above has to be a Kronecker sum, but it would be even better to do it for the regular sum. Note also that I accept any $k$, including $k = 1$, in the definition of $f$.

Does anyone know if a function like that exists?

Thanks in advance!

Gauss
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  • Some matrices have logarithms but unfortunately (like complex numbers) they may have more than one. – Henry Apr 04 '24 at 00:10
  • You will have to rethink domains for the second case, since if $M$ is $a\times b$ and $N$ is $c\times d$ then the Kronecker product $M\otimes N$ is $ac\times bd$. – coiso Apr 04 '24 at 00:31
  • Related: https://math.stackexchange.com/questions/158557/do-multiplicative-maps-of-matrices-factor-through-determinants – coiso Apr 04 '24 at 00:35
  • @coiso I realized that only after the first answer was already posted… I’ll try to think of a way to do this. Perhaps the union of all matrix rings (which isn’t a ring) could do – Gauss Apr 04 '24 at 00:36

1 Answers1

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This might be trivial, but according to Cauchy–Binet formula:

$$ \det(AB) = \det(A) \det(B) $$

and if you take the logarithm

$$ \log \det(AB) = \log \det(A) + \log \det(B) $$

suggesting that $ f(M) = \log \det (M) $ might be the function you are looking for.

zakk
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    In that same vein, then, since $\operatorname{tr}(A\otimes B) = \operatorname{tr}(A) \operatorname{tr}(B)$, the same kind of function can be found for the Kronecker product. I wonder if these are all such examples... – Gauss Apr 03 '24 at 17:28
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    $\tt+1;$And the closely related $\def\T#1{\operatorname{tr}(\log(#1))},\T{AB}=\T{A}+\T{B};$ – greg Apr 05 '24 at 23:09
  • $\def\T#1{\log\left(\operatorname{tr}\left(#1\right)\right)}\ldots$ and for the Kronecker product, just swap the log and trace functions, i.e. $$\T{A\otimes B}=\T{A} + \T{A}$$ – greg Apr 13 '24 at 18:51