# A MatrixExp question: simplifying $\int_0^t e^{A(t-t')} e^{A^T (t-t')} dt'$ for a real matrix A

I am interested in computing the following integral of a matrix exponential.

$$\int_0^t e^{A(t-t')} e^{A^T (t-t')} dt'$$

The only assumption is that $A_{n\times n}$ is real. This is simple (albeit cumbersome) to compute, given a particular $A$. I was wondering, however, if there were any other steps I could take to progress the problem further a little bit further.

For instance (assuming I did not make any mistakes), if $A$ is normal then $A = U \Lambda U^H$, where $\Lambda=\text{diag}(\lambda1,\lambda2,\ldots,\lambda_n)$ contains the eigenvalues and $U$ is unitary. Then

$$\int_0^t e^{A(t-t')} e^{A^T (t-t')} dt' = U \left[\int_0^t e^{2\Lambda(t-t')} dt'\right]U^H = U \left[(2\Lambda)^{-1} (e^{2\Lambda t} - I)\right]U^H$$

However, in general $A$ is not normal.

I appreciate the support in advance.

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 A trivial remark: we can do the substitution $s=t-t'$. The trace of the integral looks like a inner product. – Davide Giraudo Jul 19 '12 at 9:46