Eigenvalue of Block matrix: Adjacency of complete bipartite Graph Let $A\in \{0,1\}^{mn \times mn}$ be the adjacency matrix of a complete bipartite graph with $m$ and $n$ vertices each, i.e. let $A$ be the matrix consisting of two blocks $A_1\in \{0,1\}^{m \times m}$ with zeros on the diagonal and 1 elsewhere, and $A_2 \in \{0,1\}^{n \times n}$ likewise.

I know that the eigenvalues of $A_1$ and $A_2$ are given $m-1, -1, \dotsc, -1$ and $n-1, -1, \dotsc, -1$, respectively. 

It seems as if there were an easy way to combine these to the eigenvalues of $A$, and I think that the eigenvalues of $A$ should be $\sqrt{mn}, 0, \dotsc, 0, - \sqrt{mn}$, but I am not sure how to get to this. 

More general: How can one compute the eigenvalues of a block matrix given the eigenvalues of the blocks. 

Thank you very much for your help.
Here is what I could find 
"The eigenvalues of a disconnected graph
are just the eigenvalues of its components and
their multiplicities are sums of the corresponding
multiplicities in each component."
Hope this helps
Here is what I could find 
"The eigenvalues of a disconnected graph
are just the eigenvalues of its components and
their multiplicities are sums of the corresponding
multiplicities in each component."
Hope this helps