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.