The matrix A below is a block diagonal matrix where each block Ai is a $\mathbb{N} ×\mathbb{N}$ matrix with known eigenvalues and has discret spectrum .
$ A= \begin{pmatrix} A1 & 0 \\ 0 & A2\end{pmatrix} $
How do I find the spectrum of the block diagonal matrix $ A$?
The determinant of a diagonal matrix per block, for example $ D = \begin{pmatrix} B & 0 \\ 0 & C\end{pmatrix} $ where $ D \in \mathcal M_n(\mathbb K), (n \in \mathbb N)$, is $det(D)= det(B)*det(C)$.
$ \forall \lambda \in \mathbb C$, $\chi_A(\lambda) = det(\lambda I_n-A) = det(\lambda I_n-A1) * det(\lambda I_n-A2)=\chi_{A2}(\lambda)*\chi_{A1}(\lambda)$
Then the eigenvalues of $A$ are those of $A1$ and $A2$.