Assume $A$ has negative eigenvalues only, but is not necessarily symmetric or even diagonalizable. Is there a positive definite symmetric matrix $M$ such that $A$ is negative definite with respect to $M$, i.e., such that $$ \langle Ax,x\rangle_M +\langle x,Ax\rangle_{M}:= x^TA^TMx+x^{\top}MAx<-cx^TMx=:\langle x,x\rangle_M $$ for some $c>0$ and all $x$
Background: We have $$ e^{At}\to0 $$ as $t\to \infty$, but the matrix norm (corresponding to the Euclidean norm) is not monotonically decaying in general. If the answer to the question is yes, then we do have $$ \|e^{At}\|\leq e^{-ct} $$ in the matrix norm corresponding to $M$.