Example4.2.7

If a \(12\times12\) matrix has the characteristic equation

\begin{equation*} (4-\lambda)^2(-5-\lambda)(1-\lambda)^7(3-\lambda)^2 = 0 \text{,} \end{equation*}

we know there are four eigenvalues \(\lambda=4,-5,1,3\text{.}\) Without more information, all we can say about the dimensions of the eigenspaces is

\begin{equation*} \begin{aligned} 1 \leq \dim E_4 \amp {}\leq{} 2 \\ 1 \leq \dim E_{-5} \amp {}\leq{} 1 \\ 1 \leq \dim E_1 \amp {}\leq{} 7 \\ 1 \leq \dim E_3 \amp {}\leq{} 2\text{.} \\ \end{aligned} \end{equation*}

We can guarantee that \(\dim E_{-5} = 1\text{,}\) but we cannot be more specific about the dimensions of the other eigenspaces.

in-context