diagonal form of a complex matrix

Question

Let us consider a complex symmetric matrix \begin{equation} A = \begin{pmatrix} a+ib & c \\ c & -a+ib \end{pmatrix} \end{equation} where the coefficients $a,b,c$ are real. I am interested in finding an orthogonal transformation $U$ s.t. \begin{equation} U^T A U =D \end{equation} where $D$ is diagonal. The Autonne-Takagi factorization applies to complex symmetric matrices and ensures the existence and unitarity of $U$, as well as the non negativity of the components of $D$. However, this result applies for general symmetric matrices, while the matrix $A$ here has a very simple structure. Is there a result which allows to simplify the calculation of $D$ and $U$ in this specific case?

Answer 1

Hint:

Write $A=ibI+J_{a,c}$ where $\begin{equation} J_{a,c} = \begin{pmatrix} a & c \\ c & -a \end{pmatrix} \end{equation}$

Note that if you find $U$ such that $U^TJ_{a,c}U=D'$ where $D'$ is diagonal you have done because $U^TAU=ibI+D'$.

Furthermore note that $J_{a,c}$ is the moltiplication of a complex number, thus you can work with it easly.