## Proof That the Conjugate of a Symmetric Matrix by an Orthogonal Matrix is Symmetric

Theorem
A matrix
$O$
is orthogonal if its inverse equals its transpose:
$O^{-1} = O^T$
.
Let
$A$
be a symmetric matrix so that
$A=A^T$
.
If a matrix
$O$
is orthogonal and a matrix
$A$
is symmetric then
$O^{-1}AO$
is symmetric.
Proof
$(O^{-1}AO)^T =O^T A^T (O^{-1})^T$
tramspose of product of matrices property
$O^T A^T (O^{-1})^T =O^T A (O^{-1})^T$
since
$A$
is symmetric
$O^T A (O^{-1})^T =O^{-1} AO$
since
$O$
is orthogonal.