A little exercice about the matrix exponential made me think abour the following interesting property which makes a link with the sqare root of a matrix. I wish you fun proving ;-)

Let A be a real invertible n×n matrix. One defines

\sqrt{A}=\{ B\in M_n(\mathbb{R})\;\mid\; B^2=A\} and

\log(A)=\{ B\in M_n(\mathbb{R})\;\mid\; \exp(B)=A\}.

Then the set \sqrt{A} is empty exactly when \log(A) is.