I have a matrix $$\mathbf{L} =\begin{bmatrix} \mathbf{I}_{n \times n} & \frac{1}{\sqrt{mn}}\cdot\mathbf{1}_{n \times m} \\ \frac{1}{\sqrt{mn}}\cdot\mathbf{1}_{m \times n} & \mathbf{I}_{m \times m} \end{bmatrix}$$
where the diagonal blocks are identity matrices and anti-diagonal blocks are all $a$s. Is there a trick to find the eigenvalues and their multiplicities?
Numerically, I constantly get $0, 1, 2$ with multiplicities $1,m+n-2, 1$. I wonder if there exist methods that can get this solution analytically.
The context of the problem is that $\mathbf{L}$ is the graph Laplacian matrix of a complete bipartite graph $K_{n,m}$ where $$\mathbf{L} = \mathbf{I}-\mathbf{D}^{1/2}\mathbf{A}\mathbf{D}^{1/2}.$$
$\mathbf{D}$ is the diagonal degree matrix and $\mathbf{A}$ is the adjacency matrix.
Thanks in advance for the help!