Hypothesis for matrix formula

Question

I am trying to justify the formula:

$${\rm det} (M) = e^{{\rm Tr}\;log M} $$

for any square matrix M.

My idea is that it is easy to check for diagonal matrices and since det, Tr, and log are continuous functions and the set of diagonal matrices is dense in the set of matrices it must be true for any matrix.

Is that enough?

@5xum: My guess is from a series expansion, assuming M positive definite. — Learning is a mess, Apr 01 '16 at 09:21
Well, what if $M$ isn't positive definite? I mean, you say you want to prove the formula for any square matrix. Well in that case, you first need to define it for any square matrix... — 5xum, Apr 01 '16 at 09:22

score 1 · Answer 1 · answered Apr 01 '16 at 09:36

Hint: here the main problem is the definition of $\log M$ that in general, if it exists, is not a single valued function. And note that since $e^A=M$ is always an invertible matrix, the logarithm $A=\log (M)$ exists only if $M$ is invertible, so you have to pose some condition on $M$.

Hypothesis for matrix formula

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