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.
Best Answer
You also ask how to construct the matrix $A$: it is the unitary matrix of eigenvectors of the Hermitian matrix $M\cdot M^{\dagger}$.
More explicitly: The masses $m_n$ can be obtained from the eigenvalues of the matrix product $H=M\cdot M^{\dagger}$, where $M^{\dagger}$ denotes the complex conjugate of the transpose of $M$. The eigenvalues $h_n$ of $H$ are real and nonnegative, so you obtain a nonnegative mass $m_n=\sqrt{h_n}$. The matrix $A$ is the matrix of eigenvectors of $H$, so that $H=A\cdot{\rm diag}(h_1,h_2,\ldots)\cdot A^{\dagger}$. It is a unitary matrix, $AA^{\dagger}=1$.
You'll note that the matrix $A$ is not unique, you can always multiply it by a diagonal matrix of phase factors $A\mapsto A\cdot{\rm diag}(e^{i\phi_1},e^{i\phi_1},\ldots)$. The easiest way to account for this, is to just take any $A$ and calculate
$A^{\dagger}\cdot M\cdot(A^{\dagger})^{T}={\rm diag}(e^{i\psi_1}m_1,e^{i\psi_2}m_2,\ldots)$.
Then the required phases $\phi_n$ are obtained by $\phi_{n}=-\psi_n/2$.