22. Suppose the $n \times n$ matrix $A$ has the block upper triangular form
$A = \begin{bmatrix} A_{11} & A_{12} \\ O & A_{22} \end{bmatrix}$
where $A_{11}$ is $k \times k$ and $A_{22}$ is $(n-k) \times (n-k)$.
(a) If $\lambda$ is an eigenvalue of $A_{11}$, show that it is also an eigenvalue of $A$. (Hint: let $u$ be the corresponding eigenvector if $A_{11}$, and determine an $(n-k)$-vector $v$ such that $[u', v']'$ is an eigenvector of $A$ with eigenvalue $\lambda$.)
(b) If $\lambda$ is an eigenvalue of $A_{22}$ (but not of $A_{11}$), show that it is also an eigenvalue of $A$.
(c) If $\lambda$ is an eigenvalue of $A$ with corresponding eigenvector $[u', v']'$ where $u$ is a $k$-vector, show that $\lambda$ is an eigenvalue of $A_{11}$ with corresponding eigenvector $u$ or an eigenvalue of $A_{22}$ with corresponding eigenvector $v$.
(d) Conclude that $\lambda$ is an eigenvalue of $A$ if and only if it is an eigenvalue of either $A_{11}$ or $A_{22}$.
