0
$\begingroup$

Let $A$ be a symmetric matrix with real coefficients and let $O_1$ and $O_2$ be two real orthogonal matrices (i.e. $O_x^T O_x = I$).

What can one say about $$ O_1^T A O_1 + O_2^T A O_2 $$ ?

Is it possible to find two matrices $B$ and $C$ such that $$ O_1^T A O_1 + O_2^T A O_2 = B^{-1} A C $$ ?

It is simple to show that in general (unless $O_1^T A O_1$ and $O_2^T A O_2$ commute) $B \neq C$. I am particularly interested to the case when $A$ is a block matrix and the orthogonal matrices are permutation matrices.

$\endgroup$
  • $\begingroup$ Real. Thanks! (edited the question) $\endgroup$ – JacopoCrickets Apr 13 '18 at 22:35
1
$\begingroup$

This is not always possible. E.g. when $A=\operatorname{diag}(1,0),\ O_1=I$ and $O_2$ is the permutation matrix for a transposition, we get $O_1^TAO_1+O_2^TAO_2=I$, which cannot possibly be equal to the singular matrix $B^{-1}AC$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.