5 It is given that \(\lambda\) is an eigenvalue of the matrix \(\mathbf { A }\) with \(\mathbf { e }\) as a corresponding eigenvector, and \(\mu\) is an eigenvalue of the matrix \(\mathbf { B }\) for which \(\mathbf { e }\) is also a corresponding eigenvector.
- Show that \(\lambda + \mu\) is an eigenvalue of the matrix \(\mathbf { A } + \mathbf { B }\) with \(\mathbf { e }\) as a corresponding eigenvector.
The matrix \(\mathbf { A }\), given by
$$\mathbf { A } = \left( \begin{array} { r r r }
2 & 0 & 1
- 1 & 2 & 3
1 & 0 & 2
\end{array} \right)$$
has \(\left( \begin{array} { l } 1
2
1 \end{array} \right) , \left( \begin{array} { r } 1
4
- 1 \end{array} \right)\) and \(\left( \begin{array} { l } 0
1
0 \end{array} \right)\) as eigenvectors. - Find the corresponding eigenvalues.
The matrix \(\mathbf { B }\) has eigenvalues 4, 5 and 1 with corresponding eigenvectors \(\left( \begin{array} { l } 1
2
1 \end{array} \right) , \left( \begin{array} { r } 1
4
- 1 \end{array} \right)\) and \(\left( \begin{array} { l } 0
1
0 \end{array} \right)\) respectively. - Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( \mathbf { A } + \mathbf { B } ) ^ { 3 } = \mathbf { P D P } ^ { - 1 }\).