Chapter 4.5, Problem 20E

Consider a $\left(4\times 4\right)$ matrix $H$ of the form

$H=\left[\begin{array}{rrrr}\hfill \times & \hfill \times & \hfill \times & \hfill \times \\ \hfill a& \hfill \times & \hfill \times & \hfill \times \\ \hfill 0& \hfill b& \hfill \times & \hfill \times \\ \hfill 0& \hfill 0& \hfill c& \hfill \times \end{array}\right]\dots \left(11\right)$

In matrix $\left(11\right)$ the entries designated $\times$ may be zero or nonzero. Suppose, in matrix $\left(11\right)$, that $a$, $b$, and $c$ are nonzero. Let $\lambda$ be any eigenvalue of $H$. Show that the geometric multiplicity of $\lambda$ is equal to 1. [Hint: Verify that the rank of $H-\lambda I$ is exactly equal to 3.]