Singular value decomposition and eigen decomposition Given the singular value decomposition of a matrix $A$, what is the eigen decomposition of the following block matrix: $$\left(\begin{array}{cccc}
0 & A^{*}   \\ A & 0   
\end{array} \right)$$ where all including blocks are of size $n$ by $n.$
It is seen that this block matrix is Hermitian, so its eigen values are real, how do we use the SVD of A into this block matrix?
Thanks.

Given the SVD for $A=U \Sigma V^*$, we have $Av_i = \sigma_i u_i$ and $A^*u_i = \sigma_i v_i$.
Therefore, the column vector $(v_i, u_i)$ is an eigenvector of your block matrix with eigenvalue $\sigma_i$.
