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# Eigenvalues of Block Matrices

LINALG-KV97X4

Let $A$ and $B$ be $n\times n$ matrices and let $C$ be the $2n\times 2n$ block matrix $\begin{pmatrix} A & 0 \\\ 0 & B\end{pmatrix}$ and suppose that the distinct eigenvalues of $A$ are $\{\lambda_1,\dots,\lambda_r\}$ and those of $B$ are $\{\lambda'_1,\dots,\lambda'_s\}$. If $A$ and $B$ have no eigenvalues in common then the distinct eigenvalues of $C$ are:

A

$\{\lambda_i+\lambda'_j\}$ for $i=1,\dots,r$ and $j=1,\dots,s$

B

$\{\lambda_i+\lambda'_j\}$ for $i=1,\dots,r$ and $j=1,\dots,s$

C

$\{\lambda_i-\lambda'_j\}$ for $i=1,\dots,r$ and $j=1,\dots,s$

D

$\{\lambda_1,\dots,\lambda_r,\lambda'_1,\dots,\lambda'_s\}$