A real matrix $A$ is symmetric if and only if it is orthogonally diagonalizable (i.e. $A = PDP^{-1}$ for an orthogonal matrix $P$.) Proof and examples.

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##### A matrix is orthogonally diagonalizable if and only if it is symmetric. math.la.t.mat.diagonalizable.orthogonally

##### Definition of matrix diagonalization math.la.d.mat.diagonalization

##### Eigenvectors of a symmetric matrix with different eigenvalues are orthogonal. math.la.t.mat.symmetric.eig.orthogonal

##### The spectral theorem for symmetric matrices math.la.t.mat.symmetric.spectral