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).
 A: You can see that $M$ is symmetric (take its transpose!) so the characterization is "all real symmetric matrices" (at least for the real case). There's no surprise here, for $E = P^TP$, will have all entries nonnegative (they're the lengths of the columns of $P$), so they have square roots. Let $F$ be the matrix of square roots. Then letting $Q = F^{-1}P$, we have $$
Q^TQ = I$$, and 
$$P^T D P = (Q^T F^T) D (F Q) = Q^T (F^T D F) Q = Q^T ED Q,$$ 
which is a diagonalization of $M$ by the orthogonal matrix $Q$.  
A: If only $P^{T}P$ is to be diagonal, then all such matrices can be written as $P=U\Sigma$ where $U$ is a orthonormal matrix and $\Sigma$ is a diagonal matrix. However, more informative and useful is the case when $M$ is a normal matrix so that $MM^{T}=M^{T}M$. Then $M$ can be written in the way you are interested in. 
A: 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.
