We are given by a matrix $\left[\begin{array}{cc} 0 & A \\ A^T & 0 \end{array}\right]$ where $A$ is a real square matrix.
Is there a special name for this matrix?
I know that for an eigenpair $(\lambda,\left[\begin{array}{c} x \\ y \end{array}\right])$ there is an eigenpair $(-\lambda,\left[\begin{array}{c} x \\ -y \end{array}\right])$. Are there other spectral properties?