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I've been stumped by a certain problem, whose essence I've boiled down to a problem in linear algebra. Let $A$ be a real invertible matrix, and $Q$ be a real orthogonal matrix of the same size. My question is as follows:

Is $A + Q A Q^T$ invertible? If not, give a counterexample. If so, can we express the inverse in terms of $A^{-1}$ and $Q$?

I'm aware that if $A$ is symmetric and positive definite, then the above is true. But I am specifically concerned about cases when $A$ is just invertible.

EDIT: As noted in the comments below, Although the Woodbury matrix identity may be of some relevance, it does not solve the problem, since it assumes the invertibility of $A^{-1} + Q^T A^{-1} Q$.

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    $\begingroup$ Woodbury matrix identity might be of relevance here.$$(A+QAQ^T)^{-1} = A^{-1} - A^{-1}Q(A^{-1}+Q^TA^{-1}Q)^{-1}Q^TA^{-1}$$ $\endgroup$ – Adhvaitha Nov 14 '14 at 1:36
  • $\begingroup$ @Adhvaitha, this was suggested as an anonymous edit: "Although the Woodbury matrix identity may be of some relevance, it does not solve the problem, since it assumes the invertibility of $A^{−1}+Q^TA^{−1}Q$." $\endgroup$ – Joonas Ilmavirta Nov 14 '14 at 9:06
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Here is a simple counterexample $$ A=\pmatrix{0&1\\1&0}, \ Q = \pmatrix{0& 1\\-1&0}, $$ then $$ A+QAQ^T = A + Q^TAQ= 0. $$

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