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Section 5.4 Eigenvectors and Eigenspaces (G4)

Activity 5.4.1.

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 5.4.1.

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 5.4.2.

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 5.4.3.

Find a basis for the eigenspace for the matrix \(\left[\begin{array}{ccc} 5 & -2 & 0 & 4 \\ 6 & -2 & 1 & 5 \\ -2 & 1 & 2 & -3 \\ 4 & 5 & -3 & 6 \end{array}\right]\) associated with the eigenvalue \(1\text{.}\)

Activity 5.4.4.

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 \(2\text{.}\)

Exercises 5.4.1 Exercises

1.

Explain how to find a basis for the eigenspace associated to the eigenvalue \(-1 \) in the matrix

\begin{equation*} \left[\begin{array}{cccc} -2 & -1 & 0 & 1 \\ -2 & -4 & 4 & 6 \\ -2 & -2 & 0 & 3 \\ -1 & -2 & 3 & 3 \end{array}\right] \end{equation*}
Answer
\begin{equation*} \operatorname{RREF} \left[\begin{array}{cccc} -1 & -1 & 0 & 1 \\ -2 & -3 & 4 & 6 \\ -2 & -2 & 1 & 3 \\ -1 & -2 & 3 & 4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] \end{equation*}

A basis of the eigenspace is \(\left\{ \left[\begin{array}{c} 1 \\ 0 \\ -1 \\ 1 \end{array}\right] \right\} \text{.}\)

2.

Explain how to find a basis for the eigenspace associated to the eigenvalue \(2 \) in the matrix

\begin{equation*} \left[\begin{array}{cccc} 4 & 8 & 1 & -4 \\ 2 & 10 & -1 & -4 \\ 1 & 4 & -1 & -2 \\ -1 & -4 & -2 & 4 \end{array}\right] \end{equation*}
Answer
\begin{equation*} \operatorname{RREF} \left[\begin{array}{cccc} 2 & 8 & 1 & -4 \\ 2 & 8 & -1 & -4 \\ 1 & 4 & -3 & -2 \\ -1 & -4 & -2 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 4 & 0 & -2 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \end{equation*}

A basis of the eigenspace is \(\left\{ \left[\begin{array}{c} -4 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\} \text{.}\)

3.

Explain how to find a basis for the eigenspace associated to the eigenvalue \(1 \) in the matrix

\begin{equation*} \left[\begin{array}{cccc} 0 & -5 & -1 & -6 \\ -2 & -4 & -1 & -5 \\ -1 & -4 & 0 & -5 \\ 0 & -3 & -1 & -4 \end{array}\right] \end{equation*}
Answer
\begin{equation*} \operatorname{RREF} \left[\begin{array}{cccc} -1 & -5 & -1 & -6 \\ -2 & -5 & -1 & -5 \\ -1 & -4 & -1 & -5 \\ 0 & -3 & -1 & -5 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right] \end{equation*}

A basis of the eigenspace is \(\left\{ \left[\begin{array}{c} 1 \\ -1 \\ -2 \\ 1 \end{array}\right] \right\} \text{.}\)

4.

Explain how to find a basis for the eigenspace associated to the eigenvalue \(4 \) in the matrix

\begin{equation*} \left[\begin{array}{cccc} 4 & 0 & 0 & 0 \\ -1 & 6 & 0 & 4 \\ 0 & 0 & 4 & 0 \\ 2 & -4 & 0 & -4 \end{array}\right] \end{equation*}
Answer
\begin{equation*} \operatorname{RREF} \left[\begin{array}{cccc} 0 & 0 & 0 & 0 \\ -1 & 2 & 0 & 4 \\ 0 & 0 & 0 & 0 \\ 2 & -4 & 0 & -8 \end{array}\right] = \left[\begin{array}{cccc} 1 & -2 & 0 & -4 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \end{equation*}

A basis of the eigenspace is \(\left\{ \left[\begin{array}{c} 2 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\} \text{.}\)

5.

Explain how to find a basis for the eigenspace associated to the eigenvalue \(3 \) in the matrix

\begin{equation*} \left[\begin{array}{cccc} 4 & -2 & -1 & 5 \\ 0 & 4 & -2 & 1 \\ -1 & 2 & 5 & -6 \\ 0 & 1 & -7 & 9 \end{array}\right] \end{equation*}
Answer
\begin{equation*} \operatorname{RREF} \left[\begin{array}{cccc} 1 & -2 & -1 & 5 \\ 0 & 1 & -2 & 1 \\ -1 & 2 & 2 & -6 \\ 0 & 1 & -7 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right] \end{equation*}

A basis of the eigenspace is \(\left\{ \left[\begin{array}{c} -2 \\ 1 \\ 1 \\ 1 \end{array}\right] \right\} \text{.}\)

Additional exercises available at checkit.clontz.org.