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A matrix is orthogonally diagonalizable if and only if it is symmetric.

Created over 8 years ago, updated 10 days ago

Theorem. A real matrix $A$ is orthogonally diagonalizable if and only if it is symmetric.

Proof ($\Leftarrow$): Follows from the Spectral Theorem -- every real symmetric matrix has an orthonormal basis of eigenvectors.

Proof ($\Rightarrow$): If $A = PDP^T$ with $P$ orthogonal and $D$ diagonal, then:
$$A^T = (PDP^T)^T = PD^T P^T = PDP^T = A$$
(since $D^T = D$ for a diagonal matrix).

This theorem establishes a fundamental connection: among all real matrices, the symmetric ones are exactly those that can be diagonalized by an orthogonal change of basis.