Citation for trace of inverse matrix inequality

Question

I am looking for a source for the following "well known" inequality that I found here. The corresponding matrix properties/assumtions are also mentioned there in more detail.

$$ \mbox{Tr} \left( A^{-1} \right) \ge n^2 \, \mbox{Tr}(A)^{-1} $$

Can someone point me to a paper or book to which I can cite this inequality in my thesis? I looked up several known linear algebra books, but couldn't find it anywhere.

Answer 1

I found a reference to a general version of this inequality (inverse log-convex property). It can be found here. This specific case uses $\lambda=1$ and $\mu=0$ which leads to this simplified form of the inequality.

Answer 2

Just so you're aware, you need more assumptions for this inequality to be true. Let $A$ be a diagonal matrix with values $-1, -1/2$ on the diagonal.

Then $$ \mathrm{Tr}(A^{-1}) = -3 < -8/3 = 2^2 \mathrm{Tr}(A)^{-1}. $$

For $A$ with positive eigenvalues, the inequality is true and can be seen as a simple consequence of Cauchy-Schwarz.