TITLE: Prove that the eigenvalues of a block matrix are the combined eigenvalues of its blocks
QUESTION [38 upvotes]: Let $A$ be a block upper triangular matrix:
$$A = \begin{pmatrix} A_{1,1}&A_{1,2}\\ 0&A_{2,2} \end{pmatrix}$$
where $A_{1,1} ∈ C^{p \times p}$, $A_{2,2} ∈ C^{(n-p) \times (n-p)}$. Show that the eigenvalues of $A$ are the combined eigenvalues of $A_{1,1}$ and $A_{2,2}$

I've been pretty much stuck looking at this for a good hour and a half, so any help would be much appreciated. Thanks.

REPLY [2 votes]: For another approach for a proof you can use the Gershgorin disc theorem (sometimes Hirschhorn due to pronounciation differences between alphabets) to prove the disks for the individual matrices are the same as the discs for the large matrix so the sets of possible eigenvalues must be the same. This is because the radial contribution to the disks are 0 all over all entries for the lower left block since $|0| = 0$ and $0+0=0$.