TITLE: Diagonalizing a Complex Symmetric Matrix
QUESTION [16 upvotes]: 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.

REPLY [3 votes]: Since no one said it, without the condition of question 2 this can always be done. Choose a vector $v$ such that $v^T M v\neq 0$. If this can't be done then the matrix is symmetric and symplectic, hence zero, hence diagonal. If it can, scale $v$ so the product is positive real, restrict to the kernel of $v^T M$, and apply induction.