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

TheoremA 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.