I'm trying to compute
$$I(t) = \int_0^t e^{ A \tau} e^{A^T \tau} \ d \tau $$
where $A$ is a real matrix, and $A^T$ its transpose.
I know that if $A$ was symmetric and $B = A + A^T$ nonsingular, I could use the rule for the Integral of matrix exponential and the result would be
$$ I(t) = B^{-1} \left( e^{t B } - I \right) $$
What if $A$ is not symmetric, though?
You must do a numerical calculation. Let $Z(t)=\int_0^t e^{\tau A}e^{\tau A^T}d\tau$.
For example, $Z(t)$ is the solution of the ODE: $AZ(t)+Z(t)A^T=Z'(t)-I_n$, s.t. $Z(0)=0$.