Activity 4.2.4
Identify the eigenvalues, and their multiplicities, of an \(n\times n\) matrix whose characteristic polynomial is \((2\lambda)^3(3\lambda)^{10}(5\lambda)\text{.}\) What can you conclude about the dimensions of the eigenspaces? What is the dimension of the matrix? Do you have enough information to guarantee that there is a basis of \(\real^n\) consisting of eigenvectors?
Find the eigenvalues of \(\left[\begin{array}{rr} 0 \amp 1 \\ 4 \amp 4 \\ \end{array}\right]\) and state their multiplicities. Can you find a basis of \(\real^2\) consisting of eigenvectors of this matrix?

Consider the matrix \(A = \left[\begin{array}{rrr} 1 \amp 0 \amp 2 \\ 2 \amp 2 \amp 4 \\ 0 \amp 0 \amp 2 \\ \end{array}\right]\) whose characteristic equation is
\begin{equation*} (2\lambda)^2(1\lambda) = 0\text{.} \end{equation*}Identify the eigenvalues and their multiplicities.
For each eigenvalue \(\lambda\text{,}\) find a basis of the eigenspace \(E_\lambda\) and state its dimension.
Is there a basis of \(\real^3\) consisting of eigenvectors of \(A\text{?}\)

Now consider the matrix \(A = \left[\begin{array}{rrr} 5 \amp 2 \amp 6 \\ 2 \amp 2 \amp 4 \\ 2 \amp 1 \amp 2 \\ \end{array}\right]\) whose characteristic equation is also
\begin{equation*} (2\lambda)^2(1\lambda) = 0\text{.} \end{equation*}Identify the eigenvalues and their multiplicities.
For each eigenvalue \(\lambda\text{,}\) find a basis of the eigenspace \(E_\lambda\) and state its dimension.
Is there a basis of \(\real^3\) consisting of eigenvectors of \(A\text{?}\)

Consider the matrix \(A = \left[\begin{array}{rrr} 5 \amp 2 \amp 6 \\ 4 \amp 1 \amp 8 \\ 2 \amp 1 \amp 2 \\ \end{array}\right]\) whose characteristic equation is
\begin{equation*} (2\lambda)(1\lambda)(1\lambda) = 0\text{.} \end{equation*}Identify the eigenvalues and their multiplicities.
For each eigenvalue \(\lambda\text{,}\) find a basis of the eigenspace \(E_\lambda\) and state its dimension.
Is there a basis of \(\real^3\) consisting of eigenvectors of \(A\text{?}\)