Proof of log identity for positive definite matrices

Question

On Wikipedia, it is claimed that $\rm{tr}(\log(AB)) = \rm{tr}\log(A) + \rm{tr}\log(B)$.

The result is valid only if $A$ and $B$ are positive definite.

I use this result in entropy calculations but I do not know how to prove it - in particular to show that it only works for positive definite matrices. Could someone help me with the proof?

Answer 1

The spectra of $A,B,AB$ are included in $(0,+\infty)$. Then, it suffices to prove the equality of the exponentials of RHS and LHS. Since there are no $\leq 0$ eigenvalues, we use the principal logarithm.

Let $U$ be symmetric $>0$.

$\exp(tr(\log(U))=\det(\exp(\log(U)))=\det(U)$.

We conclude with $\det(AB)=\det(A)\det(B)$.

Remark. i) If ,$A,B$ are symmetric but with $<0$ eigenvalues, then we cannot use the principal logarithm.

ii) if $A,B$ are invertible but not symmetric and have no $<0$ eigenvalues, then $AB$ may have $<0$ eigenvalues

example $A=B=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$.