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.

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