Symmetric matrices and exponentials

Question

My question is simple, suppose I have a matrix $A$, how do I know that it is an exponential of symmetric matrix $S$, i.e what are the conditions on $A$ such that $A= \exp(S)$ where $S$ is symmetric

Answer 1

We claim that $A = e^S$ for $S$ symmetric if and only if $A$ is symmetric and has positive eigenvalues.

If $A = e^S$, $A$ must be symmetric as well since $(\text{exp(S)})^T = \text{exp}(S^T)$. To show positivity of eigenvalues, let $v$ be an eigenvector of $S$, with eigenvalue $\lambda$. Then $Av = e^S v = e^{\lambda} v$, and $e^{\lambda} > 0.$

Now assume $A$ is symmetric and has positive eigenvalues. Then there exists an orthogonal matrix $O$ such that $O^T A O = D$, where $D$ is diagonal and has positive entries on the diagonal. Then there exists a diagonal matrix $M$ such that $D = e^M$. Thus $A = O e^M O^T = \text{exp}(OMO^T)$ and clearly $OMO^T$ is symmetric.