log of determinant of PD matrix

Question

I found in wikipedia a following inequality:

$ tr(I-A^{-1})\le\log(\det A)\le tr(A-I) $

When $A$ is positive definite matrix.

Why is it true?

Answer 1

You can reduce the proof to the case when $A$ is diagonal. In that case the inequality becomes $\sum (1-\frac 1 {\lambda_i}) \leq \sum \log \lambda_i \leq \sum (\lambda_i -1)$ and this follows from the inequalities $1 -\frac 1 x \leq \log x \leq x-1$ valid for all $x >0$. [For the left hand inequality note that $\log x-1 +\frac 1 x$ attains its minimum at $x=1$].