Let $A \in \Bbb R ^{n \times p}$ be a real-valued matrix. What is the necessary and sufficient condition on $A$ such that the matrix exponential $$\exp( - tA^TA)$$ goes to zero as $t$ approaches infinity?
When $n > p$, the matrix exponential goes to zero but when $n < p$, the matrix exponential is not zero. I am randomly generating $A$ from normal random generator in MATLAB and then compute matrix exponential.
$e^{-tB}\to 0$ as $t\to\infty$ if and only if all the (complex) eigenvalues of $B$ have strictly positive real part. $A^TA$ is symmetric positive semidefinite, therefore all its eigenvalues are real and $\ge 0$. Also, $\ker (A^TA)=\ker A$, meaning that $0$ isn't an eigenvalue of $A^TA$ (id est, that $e^{-tA^TA}\to 0$ as $t\to\infty$) if and only if $\ker A=\{0\}$.