Characterization of Matrices Diagonalizable by Matrices P such that P times P^Transpose is Diagonal Let $M$ be a square matrix with complex entries.
What is a characterization of $M$ such that $M = P^{T} D P$, where both $D$ and $P^{T} P$ are diagonal matrices?
For example, such a characterization includes all real symmetric matrices using only orthogonal matrices for $P$ (so that $P^{T} P$ is the identity matrix, which of course is diagonal).

It is a theorem due to Takagi that any complex (entrywise) symmetric matrix may be written as $M=PDP^T$, and $P$ may be chosen to be unitary.
As others have noted, it is clear that only symmetric matrices can be factored in this way, so that symmetry is both sufficient and necessary.
