Under what conditions can I orthogonally diagonalize a matrix

Question

I know that you can orthogonally diagonalize a matrix if it's symmetric. Under what other conditions can I orthogonally diagonalize a matrix? And if a matrix is diagonalizable, is it orthogonally diagonalizable?

Answer 1

Any matrix that is orthogonally diagonalizable (in the conventional usage of this term) is necessarily symmetric. Indeed, if $A = UDU^T$ for an orthogonal matrix $U$ and a real, diagonal matrix $D$, then $$ A^T = (UDU^T)^T = U^{TT}D^TU^T = UDU^T = A. $$ A matrix that is diagonalizable will not necessarily be orthogonally diagonalizable. For instance, the matrix $$ A = \pmatrix{1&2\\0&3} $$ is diagonalizable, as we could deduce from the fact that it $2 \times 2$ with $2$ distinct, real eigenvalues. However, since $A$ is not symmetric, it cannot be orthogonally diagonalizable.