Evaluation of exponential matrix integral

Question

Let $A$ and $B$ be symmetric $n\times n$ matrices, and let $D$ be a diagonal $n\times n$ matrix. Assume that all matrices are invertible. In my particular case of interest $A=B$ but this may be irrelevant. The question is to evaluate the indefinite matrix integral

$$I \equiv \int dt \; e^{-t D\cdot A} \cdot B \cdot e^{-t A \cdot D}$$

which arises from an instanton problem. It looks simple, and yet ...

Can’t you just evaluate $B\cdot \int e^{-c t}dt,c=D\cdot A+A\cdot D$? — Тyma Gaidash, Jul 13 '23 at 02:52
$B$ does not commute with either of the exponentials, in general — eric, Jul 13 '23 at 03:16

score 0 · Answer 1 · answered Jul 13 '23 at 07:05

0

With $C=AD,$ apparently nothing simpler than, up to a symmetric matrix constant $$-\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\frac{(-t)^{n+m+1}}{(n+m+1)n!m!}(C^T)^nBC^m$$

answered Jul 13 '23 at 07:05

Gérard Letac

4,357

Evaluation of exponential matrix integral

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