Eigenvectors and Eigenspaces (G4)

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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 5.4.2.

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 \(\lambda\text{.}\)

Activity 5.4.3.

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 \(3\text{.}\)

Activity 5.4.4.

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 \(1\text{.}\)

Activity 5.4.5.

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 \(2\text{.}\)

Subsection 5.4.1 Videos

Figure 5.4.6. Video: Finding eigenvectors

Exercises 5.4.2 Exercises

Exercises available at checkit.clontz.org ¹.

https://checkit.clontz.org/#/banks/tbil-la/G4/