For a symmetric matrix M with complex entries, I want to diagonalize it using a matrix A, such that
$AMA^T = D$, where D is a diagonal matrix with real-positive entries.
Question 1: When can this be done?
Question 2: Is $A$ unitary, i.e., is $A^\dagger A = 1$ ?
Question 3: How do I construct $A$?
The question is motivated by Majorana masses of fermions, which are complex symmetric matrices, and need to be diagonalized as above to get the physical masses. Obviously masses need to be positive and the basis-rotation by $A$ must preserve probabilities, and needs to be unitary.