Determinant of a specific block matrix with 0 matrix in diagonals.

Question

I am trying to determine the eigenvalues of a block matrix of the form

$$\begin{pmatrix} O & B \\ C & O \end{pmatrix}$$ where $O,B$ and $C$ are $n\times n$ matrices. How would I go about this, given that $B = kI$ for some constant $k$ and that $C$ is a symmetric matrix whose eigenvalues are known?

Answer 1

The block vector $\pmatrix{u\cr v\cr}$ is an eigenvector for eigenvalue $\lambda$ if $$ \eqalign{B v &= \lambda u\cr C u &= \lambda v\cr}$$ which (if $\lambda \ne 0$) is equivalent to $$\eqalign{CB v &= \lambda^2 v\cr u &= \lambda^{-1} B v}$$ I'll leave the case $\lambda=0$ to you. So basically you take the square roots (both of them) of the eigenvalues of $CB$.