Lesson 8 Eigenvalues and Eigenvectors

8.1 Lecture Content

Video: Eigenvalues and Eigenvectors

Video: Eigenvalues and Eigenvectors

8.2 Lecture Notes

8.2.1 Eigenvalues and Eigenvectors

Let \(A\) be a square matrix. A non-zero vector \(\vec{v}\) is an eigenvector of \(A\) if: \[ A\vec{v} = \lambda \vec{v} \] for some scalar \(\lambda\). The scalar \(\lambda\) is called the eigenvalue corresponding to the eigenvector \(\vec{v}\).

  • An eigenvector for \(A\) will have the same direction once it is multiplied by \(A\).
  • If the eigenvector changes length, it does so by a factor of the eigenvalue.

8.2.2 Finding Eigenvalues (for \(2 \times 2\) or \(3 \times 3\) matrices)

  1. Start with the matrix equation \(A\vec{v} = \lambda\vec{v}\).
  2. Rearrange to \((A - \lambda I)\vec{v} = 0\).
  3. Compute the characteristic polynomial by solving: \[ \det(A - \lambda I) = 0 \]
  4. The roots \(\lambda\) of this polynomial are the eigenvalues of \(A\).

8.2.3 Finding Eigenvectors

Once you have the eigenvalues:

  1. For each eigenvalue \(\lambda\), plug it into \(A - \lambda I\).
  2. Solve the homogeneous system: \[ (A - \lambda I)\vec{v} = 0 \]
  3. The solution set (null space) gives the eigenvectors.
  4. Any basis for this null space forms a basis for the eigenspace corresponding to \(\lambda\).

8.2.4 Diagonalization (for \(2 \times 2\) matrices with two distinct eigenvalues)

A matrix \(A\) is diagonalizable if there exists an invertible matrix \(P\) and a diagonal matrix \(D\) such that: \[ A = PDP^{-1} \]

To diagonalize:

  1. Find the eigenvalues \(\lambda_1, \lambda_2\).
  2. For each \(\lambda\), find the corresponding eigenvectors.
  3. Form matrix \(P\) using the eigenvectors as columns.
  4. Construct diagonal matrix \(D\) with the eigenvalues on the diagonal.
  5. Confirm that \(A = PDP^{-1}\).

Diagonalization simplifies matrix powers and exponentials, as: \[ A^n = P D^n P^{-1} \] when \(A\) is diagonalizable.

8.3 Examples

Video: Solutions

  1. Find the eigenvalues of the matrix
    \[ \begin{bmatrix} 7 & 3 \\ -2 & 2 \end{bmatrix} \]

  2. One of the eigenvalues of the matrix
    \[ \begin{bmatrix} 21 & -80 \\ 4 & -15 \end{bmatrix} \]
    is 8. Find an associated eigenvector.

  3. Suppose matrix \(A\) is a \(3 \times 4\) matrix such that
    \[ A \cdot \begin{bmatrix} -4 \\ 6 \\ 4 \end{bmatrix} = \begin{bmatrix} -2 \\ 3 \\ 2 \end{bmatrix}. \]

Find an eigenvalue of \(A\).

4.Let
\[ M = \begin{bmatrix} 1 & 2 \\ -1 & 4 \end{bmatrix}. \]

Find an eigenvalue and a corresponding eigenvector..

  1. Find a matrix \(A\) with eigenvalues 1 and \(-2\). Also find the corresponding eigenvectors.

  2. Suppose \(A\) is a matrix and
    \[ u = \begin{bmatrix} 5 \\ -1 \\ 1 \\ 4 \end{bmatrix} \] is an eigenvector corresponding to \(\lambda = -3\), and
    \[ v = \begin{bmatrix} -1 \\ 2 \\ 1 \\ 4 \end{bmatrix} \] is an eigenvector corresponding to \(\lambda = 2\).

Find the following:

\[ A u, \quad A v, \quad A (u + v) \]

8.4 Practice Problems

Practice the techniques discussed in class and in the online videos by solving the following examples.

  1. If \[ A = \begin{bmatrix}3&-2&2\\1&2&1\\0&2&1 \end{bmatrix} \] determine which of the following are eigenvectors of \(A\) and what their associated eigenvalue is.
    1. \(\begin{bmatrix}2 \\ -1\\ -2 \end{bmatrix}\)
    2. \(\begin{bmatrix}0\\1\\1 \end{bmatrix}\)
    3. \(\begin{bmatrix} 1\\1\\1 \end{bmatrix}\)
    4. \(\begin{bmatrix}0\\0\\0 \end{bmatrix}\)
  2. Find the eigenvalues of \[ A = \begin{bmatrix}-5 & 2 \\ -7 & 4 \end{bmatrix}. \] Sketch the eigenvectors before and after left-sided multiplication by \(A\).
  3. Find the eigenvalues of \[ A = \begin{bmatrix}1&2&4\\0&4&7\\0&0&6 \end{bmatrix}. \]
  4. Find the eigenvalues of \[ A = \begin{bmatrix}2&2&-2\\1&3&-1\\-1&1&1 \end{bmatrix}. \]
  5. Show that the eigenvalues of \[ A = \begin{bmatrix}-5 & 2 \\ -7 & 4 \end{bmatrix} \] are \(\lambda = \pm 2\), then determine a basis for each eigenspace of \(A\).
  6. (Applied) We can decompose a matrix \(A\) into the form \(A = PDP^{-1}\) where \(D\) is a matrix of eigenvalues on the main diagonal and \(P\) is a matrix of eigenvectors. Compute \(P\) and \(D\) for \[ A = \begin{bmatrix}3 & 0 \\ 0 & -2\end{bmatrix}. \] Compute \(A^4\).
  7. (Applied) We can decompose a matrix \(A\) into the form \(A = PDP^{-1}\) where \(D\) is a matrix of eigenvalues on the main diagonal and \(P\) is a matrix of eigenvectors. Compute \(P\) and \(D\) for \[ A = \begin{bmatrix}-1 &4 \\ 0 & 3\end{bmatrix}. \] Compute \(A^3\).

8.5 Self-Assessment

Time yourself and try to solve the following questions within twenty minutes.

  1. Find the eigenvalues of \[ A = \begin{bmatrix}-3&0\\0&4 \end{bmatrix}. \]

  2. Find the eigenvalues of \[ A = \begin{bmatrix}1&4\\3&2 \end{bmatrix}. \] Sketch the eigenvectors before and after left-sided multiplication by \(A\).

  3. Find the eigenvalues of \[ A = \begin{bmatrix}5&-4&4\\2&-1&2\\0&0&2\end{bmatrix}. \]

  4. We can decompose a matrix \(A\) into the form \(A = PDP^{-1}\) where \(D\) is a matrix of eigenvalues on the main diagonal and \(P\) is a matrix of eigenvectors. Compute \(P\) and \(D\) for \[ A = \begin{bmatrix}2 &3\\3 & 2\end{bmatrix}. \] Compute \(A^{10}\).

8.6 Lesson Checklist

This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an \(X\) in the appropriate box beside the skill below.

\[\begin{align*} &\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\ &\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\ &\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.} \end{align*}\]

Skill D CON COM
Find all eigenvalues of a \(2 \times 2\) matrix.
Find all eigenvalues of a \(3 \times 3\) matrix.
Find the eigenvectors of a \(2 \times 2\) matrix.