Can we find the eigenvalues of this block matrix?

Question

Suppose that $M$ is an $m \times m$ matrix. Can we find the eigenvalues of the following block matrix?

$$B = \begin{bmatrix} -I & M\\ M^T & -I\end{bmatrix}$$

in terms of the eigenvalues of $M$?

Answer 1

You can relate the eigenvalues of $B$ to those of $M^TM$. First of all, we have $$ (B-\lambda)\binom x y = \binom 0 0\quad\Longleftrightarrow\quad My = (1+\lambda)x\quad\text{and}\quad M^Tx = (1+\lambda)y. $$ From here, it is quite elementary to show that $\lambda$ is an eigenvalue of $B$ if and only if $(1+\lambda)^2$ is an eigenvalue of $M^TM$. This also shows you that if $B$ has an eigenvalue $\lambda$, then also $-2-\lambda$ is an eigenvalue of $B$. That is, the eigenvalues of $B$ are symmetric with respect to the point $-1$.

Even if $M$ is square, you can in general not relate the eigenvalues of $B$ to those of $M$ (except the zero eigenvalue). Only if $M$ is symmetric and positive semi-definite, then $\lambda$ is an eigenvalue of $B$ if and only if $|1+\lambda|$ (i.e., the distance of $\lambda$ to $-1$) is an eigenvalue of $M$.