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My goal is to express the result of the following integral in closed-form (using for example traces of the matrices):

$$ \int_{-\infty}^{\infty} b^T G(x)^T H G(x) b dx $$ where $$ b \in \mathbb{R}^{m \times 1} $$ $$ H \in \mathbb{R}^{m \times m} $$ $$ G = (xI - H)^{-1} $$

Taiben
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  • Did you try to solve in depth the case $n=1$? The result is already enlightening for the general case... – Did Sep 03 '14 at 08:32
  • Yes I did. I made an edit to remove the unnecessary terms. – Taiben Sep 03 '14 at 14:37
  • True, $a^\top a$ was unnecessary, but my point was that you should show your computations in the $n=1$ case, and the result you arrived at in this case. – Did Sep 03 '14 at 14:40

1 Answers1

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You may decompose the non-Hermitian operator $H$:

$$ H = \sum_m h_m \vert m \rangle \langle \tilde{m} \vert $$

and express the result in terms of the pseudoeigenvalues.

user20494
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