Chapter 3.P1, Problem 3P

Computing the Entries of $K$ from Its Eigenvalues and Eigenvectors. Assume that the eigenvalues ${\lambda }_1$ and ${\lambda }_2$ and corresponding eigenvectors ${\text{v}}_1$ and ${\text{v}}_2$ of $K$ are known. Show that the entries of the matrix $K$ must satisfy the following systems of equations:

$\left(\begin{array}{cc}{v}_{11}& v_{21}\\ v_{12}& v_{22}\end{array}\right)\left(\begin{array}{c}{k}_{11}\\ k_{12}\end{array}\right)=\left(\begin{array}{c}{\lambda }_1{v}_{11}\\ {\lambda }_2{v}_{12}\end{array}\right)$ (iii)

and

$\left(\begin{array}{cc}{v}_{11}& v_{21}\\ v_{12}& v_{22}\end{array}\right)\left(\begin{array}{c}{k}_{21}\\ k_{22}\end{array}\right)=\left(\begin{array}{c}{\lambda }_1{v}_{21}\\ {\lambda }_2{v}_{22}\end{array}\right)$(iv)

or, using matrix notation, $KV=V\Lambda$, where

$V=\left(\begin{array}{cc}{v}_{11}& v_{21}\\ v_{12}& v_{22}\end{array}\right)$ and $\Lambda =\left(\begin{array}{cc}{\lambda }_1& 0\\ 0& {\lambda }_2\end{array}\right)$

$K=\left(\begin{array}{cc}{K}_{11}& K_{12}\\ K_{21}& K_{22}\end{array}\right)$