to find all the eigenvectors \(\vec x\) such that \(A\vec x=-2\vec x\text{.}\)
Definition5.4.2.
Since the kernel of a linear map is a subspace of \(\IR^n\text{,}\) and the kernel obtained from \(A-\lambda I\) contains all the eigenvectors associated with \(\lambda\text{,}\) we call this kernel the eigenspace of \(A\) associated with \(\lambda\text{.}\)
Activity5.4.3.
Find a basis for the eigenspace for the matrix \(\left[\begin{array}{ccc}
0 & 0 & 3 \\ 1 & 0 & -1 \\ 0 & 1 & 3
\end{array}\right]\) associated with the eigenvalue \(3\text{.}\)
Activity5.4.4.
Find a basis for the eigenspace for the matrix \(\left[\begin{array}{cccc}
5 & -2 & 0 & 4 \\ 6 & -2 & 1 & 5 \\ -2 & 1 & 2 & -3 \\ 4 & 5 & -3 & 6
\end{array}\right]\) associated with the eigenvalue \(1\text{.}\)
Activity5.4.5.
Find a basis for the eigenspace for the matrix \(\left[\begin{array}{cccc}
4 & 3 & 0 & 0 \\ 3 & 3 & 0 & 0 \\ 0 & 0 & 2 & 5 \\ 0 & 0 & 0 & 2
\end{array}\right]\) associated with the eigenvalue \(2\text{.}\)
Given a matrix \(A\text{,}\) let \(\{\vec{v_1},\vec{v_2},\ldots,\vec{v_n}\}\) be the eigenvectors with associated distinct eigenvalues \(\{\lambda_1,\lambda_2,\ldots, \lambda_n\}\text{.}\) Prove the set of eigenvectors is linearly independent.