Learning Outcomes
- Find a basis for the eigenspace of a \(4\times 4\) matrix associated with a given eigenvalue.
Section 5.4 Eigenvectors and Eigenspaces (GT4)
Subsection 5.4.1 Warm Up
Activity 5.4.1.
Which of the following vectors is an eigenvector for \(A=\left[\begin{array}{cccc} 2 & 4 & -1 & -5 \\ 0 & 0 & -3 & -9 \\ 1 & 1 & 0 & 2 \\ -2 & -2 & 3 & 5 \end{array}\right]\text{?}\)
- \(\displaystyle \left[\begin{matrix}-2\\1\\0\\1\end{matrix}\right]\)
- \(\displaystyle \left[\begin{matrix}-3\\3\\-2\\1\end{matrix}\right]\)
Subsection 5.4.2 Class Activities
Activity 5.4.2.
It’s possible to show that \(-2\) is an eigenvalue for \(\left[\begin{array}{ccc}-1&4&-2\\2&-7&9\\3&0&4\end{array}\right]\text{.}\)
Compute the kernel of the transformation with standard matrix
\begin{equation*}
A-(-2)I
=
\left[\begin{array}{ccc} \unknown & 4&-2 \\ 2 & \unknown & 9\\3&0&\unknown \end{array}\right]
\end{equation*}
to find all the eigenvectors \(\vec x\) such that \(A\vec x=-2\vec x\text{.}\)
Definition 5.4.3.
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 \(\lambda\text{.}\)
Activity 5.4.4.
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 \(3\text{.}\)
Activity 5.4.5.
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 \(1\text{.}\)
Activity 5.4.6.
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 \(2\text{.}\)
Subsection 5.4.3 Individual Practice
Activity 5.4.7.
Suppose that \(T\colon\IR^2\to\IR^2\) is a linear transformation with standard matrix \(A\text{.}\) Further, suppose that we know that \(\vec{u}=\left[\begin{matrix}1\\-1\end{matrix}\right]\) and \(\vec{v}=\left[\begin{matrix}2\\-3\end{matrix}\right]\) are eigenvectors corresponding to eigenvalues \(2\) and \(-3\) respectively.
(a)
Express the vector \(\vec{w}=\left[\begin{matrix}2\\1\end{matrix}\right]\) as a linear combination of \(\vec{u},\vec{v}\text{.}\)
(b)
Determine \(T(\vec{w})\text{.}\)
Subsection 5.4.4 Videos
Exercises 5.4.5 Exercises
Exercises available at
https://tbil.org/linear-algebra/preview/exercises/#/bank/GT4/.Subsection 5.4.6 Mathematical Writing Explorations
Exploration 5.4.8.
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.Subsection 5.4.7 Sample Problem and Solution
Sample problem Example C.1.25.
