Eigenvalues and Eigenvectors

Exercise 1

Confirm by multiplication that x is an eigenvector of A, and find the corresponding eigenvalue.

  1. \( A = \begin{bmatrix} 1 & 2 \\ 3 & 2 \end{bmatrix}, \quad x = \begin{bmatrix} 1 \\ -1 \end{bmatrix} \)
  2. \( A = \begin{bmatrix} 5 & -1 \\ 3 & 2 \end{bmatrix}, \quad x = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \)
  3. \( A = \begin{bmatrix} 4 & 0 & 0 \\ 2 & 3 & 2 \\ 1 & 0 & 4 \end{bmatrix}, \quad x = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \)
  4. \( A = \begin{bmatrix} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{bmatrix}, \quad x = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \)

Exercise 2

Find the characteristic equation, the eigenvalues, and bases for the eigenspaces of the matrix.

  1. \( \begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix} \)
  2. \( \begin{bmatrix} -2 & -1 \\ 2 & 3 \end{bmatrix} \)
  3. \( \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)
  4. \( \begin{bmatrix} 1 & -2 \\ 0 & 1 \end{bmatrix} \)
  5. \( \begin{bmatrix} 1 & 2 \\ 2 & 0 \end{bmatrix} \)
  6. \( \begin{bmatrix} 0 & 2 \\ 2 & -1 \end{bmatrix} \)
  7. \( \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} \)
  8. \( \begin{bmatrix} -2 & -1 \\ 2 & -1 \end{bmatrix} \)
  9. \( \begin{bmatrix} 4 & 0 & 1 \\ -2 & 1 & 0 \\ 2 & 0 & 1 \end{bmatrix} \)
  10. \( \begin{bmatrix} 1 & 0 & -2 \\ 0 & -2 & 0 \\ 0 & -2 & 4 \end{bmatrix} \)
  11. \( \begin{bmatrix} 6 & 3 & -8 \\ 0 & -2 & 0 \\ 1 & 0 & -3 \end{bmatrix} \)
  12. \( \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix} \)
  13. \( \begin{bmatrix} 4 & 0 & -1 \\ 0 & 3 & 2 \\ 2 & 0 & 1 \end{bmatrix} \)
  14. \( \begin{bmatrix} 1 & -3 & -3 \\ 6 & -6 & 4 \\ 6 & -6 & -4 \end{bmatrix} \)

Exercise 3

Find the characteristic equation of the matrix by inspection.

  1. \( \begin{bmatrix} 3 & 0 & 0 \\ -2 & 7 & 0 \\ 4 & 8 & 1 \end{bmatrix} \)
  2. \( \begin{bmatrix} 9 & -8 & 6 \\ -1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 7 \end{bmatrix} \)

Exercise 4

Find the eigenvalues and a basis for each eigenspace of the linear operator defined by the stated formula.

  1. \( T(x, y) = (x + 4y, 2x + 3y) \)
  2. \( T(x, y, z) = (2x - y - z, x - z, -x + y + 2z) \)

Exercise 5

Let \( D^2: C^\infty(-\infty, \infty) \to C^\infty(-\infty, \infty) \) be the operator that maps a function into its second derivative.

  1. Show that \( D^2 \) is linear.
  2. Show that if \( \omega \) is a positive constant, then \( \sin(\sqrt{\omega}x) \) and \( \cos(\sqrt{\omega}x) \) are eigenvectors of \( D^2 \), and find their corresponding eigenvalues.

Exercise 6

Let \( D^2: C^\infty \to C^\infty \) be the linear operator in Exercise 5. Show that if \( \omega \) is a positive constant, then \( \sinh(\sqrt{\omega}x) \) and \( \cosh(\sqrt{\omega}x) \) are eigenvectors of \( D^2 \), and find their corresponding eigenvalues.

Exercise 7

Find the eigenvalues and the corresponding eigenspaces of the stated matrix operator on \( \mathbb{R}^2 \). No computations are needed.

  1. Reflection about the line \( y = x \).
  2. Orthogonal projection onto the \( x \)-axis.
  3. Rotation about the origin through a positive angle of 90°.
  4. Contraction with factor \( k \) ( \( 0 \leq k < 1 \) ).
  5. Shear in the \( x \)-direction by a factor \( k \) ( \( k \neq 0 \) ).

Exercise 8

Find the eigenvalues and the corresponding eigenspaces of the stated matrix operator on \( \mathbb{R}^2 \). No computations are needed.

  1. Reflection about the \( y \)-axis.
  2. Rotation about the origin through a positive angle of 180°.
  3. Dilation with factor \( k \) ( \( k > 1 \) ).
  4. Expansion in the \( y \)-direction with factor \( k \) ( \( k > 1 \) ).
  5. Shear in the \( y \)-direction by a factor \( k \) ( \( k \neq 0 \) ).

Exercise 9

Find the eigenvalues and corresponding eigenspaces of the stated matrix operator on \( \mathbb{R}^3 \). No computations are needed.

  1. Reflection about the \( xy \)-plane.
  2. Orthogonal projection onto the \( xz \)-plane.
  3. Counterclockwise rotation about the positive \( x \)-axis through an angle of 90°.
  4. Contraction with factor \( k \) ( \( 0 \leq k < 1 \) ).

Exercise 10

Find the eigenvalues and corresponding eigenspaces of the stated matrix operator on \( \mathbb{R}^3 \). No computations are needed.

  1. Reflection about the \( xz \)-plane.
  2. Orthogonal projection onto the \( yz \)-plane.
  3. Counterclockwise rotation about the positive \( y \)-axis through an angle of 180°.
  4. Dilation with factor \( k \) ( \( k > 1 \) ).

Exercise 11

Let A be a \(2 \times 2\) matrix, and call a line through the origin of \(\mathbb{R}^2\) invariant under A if \(Ax\) lies on the line when \(x\) does. Find equations for all lines in \(\mathbb{R}^2\), if any, that are invariant under the given matrix.

  1. \( A = \begin{bmatrix} 4 & -1 \\ 2 & 1 \end{bmatrix} \)
  2. \( A = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \)

Exercise 12

Find \(\det(A)\) given that A has \(p(\lambda)\) as its characteristic polynomial.

  1. \( p(\lambda) = \lambda^3 - 2\lambda^2 + \lambda + 5 \)
  2. \( p(\lambda) = \lambda^3 - \lambda + 7 \)

Exercise 13

Suppose that the characteristic polynomial of some matrix A is found to be \( p(\lambda) = (\lambda - 1)(\lambda - 3)^2(\lambda - 4)^3 \). In each part, answer the question and explain your reasoning.

  1. What is the size of A?
  2. Is A invertible?
  3. How many eigenspaces does A have?

Exercise 14

The eigenvectors that we have been studying are sometimes called right eigenvectors to distinguish them from left eigenvectors, which are \( 1 \times n \) row matrices \( y^T \) that satisfy the equation \( y^T A = \mu y^T \) for some scalar \(\mu\). For a given matrix, how are the right eigenvectors and their corresponding eigenvalues related to the left eigenvectors and their corresponding eigenvalues?

Exercise 15

Find a \( 3 \times 3 \) matrix A that has eigenvalues \( 1, -1, \) and \( 0 \), and for which

\[ \begin{bmatrix} -1 \\ 1 \\ 1 \end{bmatrix}, \quad \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}, \quad \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix} \]

are their corresponding eigenvectors.

Exercise 16

Prove that the characteristic equation of a \( 2 \times 2 \) matrix A can be expressed as

\[ \lambda^2 - \text{tr}(A)\lambda + \det(A) = 0 \]

where \(\text{tr}(A)\) is the trace of A.

Exercise 17

Use the result in Exercise 16 to show that if

\[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]

then the solutions of the characteristic equation of A are

\[ \lambda = \frac{1}{2} \left( (a + d) \pm \sqrt{(a - d)^2 + 4bc} \right) \]

Use this result to show that A has:

  1. Two distinct real eigenvalues if \( (a - d)^2 + 4bc > 0 \).
  2. Two repeated real eigenvalues if \( (a - d)^2 + 4bc = 0 \).
  3. Complex conjugate eigenvalues if \( (a - d)^2 + 4bc < 0 \).

Exercise 18

Let A be the matrix in Exercise 17. Show that if \( b \neq 0 \), then

\[ x_1 = \begin{bmatrix} -b \\ (a - \lambda_1) \end{bmatrix}, \quad x_2 = \begin{bmatrix} -b \\ (a - \lambda_2) \end{bmatrix} \]

are eigenvectors of A that correspond, respectively, to the eigenvalues

\[ \lambda_1 = \frac{1}{2} \left( (a + d) + \sqrt{(a - d)^2 + 4bc} \right) \]

and

\[ \lambda_2 = \frac{1}{2} \left( (a + d) - \sqrt{(a - d)^2 + 4bc} \right). \]

Exercise 19

Use the result of Exercise 17 to prove that if

\[ p(\lambda) = \lambda^2 + c_1\lambda + c_2 \]

is the characteristic polynomial of a \( 2 \times 2 \) matrix, then

\[ p(A) = A^2 + c_1 A + c_2 I = 0. \]

(Informally, A satisfies its characteristic equation. This result is true for \( n \times n \) matrices as well.)

Exercise 20

Prove:

  1. If \( a, b, c, d \) are integers such that \( a + b = c + d \), then

    \[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]

    has integer eigenvalues.

  2. If \( \lambda \) is an eigenvalue of an invertible matrix A and \( x \) is a corresponding eigenvector, then \( 1/\lambda \) is an eigenvalue of \( A^{-1} \) and \( x \) is a corresponding eigenvector.

  3. If \( \lambda \) is an eigenvalue of A, \( x \) is a corresponding eigenvector, and \( s \) is a scalar, then \( \lambda - s \) is an eigenvalue of \( A - sI \) and \( x \) is a corresponding eigenvector.

  4. If \( \lambda \) is an eigenvalue of A and \( x \) is a corresponding eigenvector, then \( s\lambda \) is an eigenvalue of \( sA \) for every scalar \( s \) and \( x \) is a corresponding eigenvector.

Exercise 21

Find the eigenvalues and bases for the eigenspaces of

\[ A = \begin{bmatrix} -2 & 3 & 3 \\ -2 & 3 & 2 \\ -4 & 2 & 5 \end{bmatrix} \]

and then use Exercises 20.2 and Exercises 20.3 to find the eigenvalues and bases for the eigenspaces of:

  1. \( A^{-1} \)
  2. \( A - 3I \)
  3. \( A + 2I \)

Exercise 22

Prove that the characteristic polynomial of an \( n \times n \) matrix A has degree \( n \) and that the coefficient of \( \lambda^n \) in that polynomial is 1.

Exercise 23

  1. Prove that if A is a square matrix, then A and \( A^T \) have the same eigenvalues.
    Hint: Look at the characteristic equation \( \det(\lambda I - A) = 0 \).

  2. Show that A and \( A^T \) need not have the same eigenspaces.
    Hint: Use the result in Exercise 18 to find a \( 2 \times 2 \) matrix for which A and \( A^T \) have different eigenspaces.

  3. Prove that the intersection of any two distinct eigenspaces of a matrix A is \(\{ 0 \}\).

Exercises 24 : True-False

Determine whether the statement is true or false, and justify your answer.

  1. If A is a square matrix and \( Ax = \lambda x \) for some nonzero scalar \(\lambda\), then \( x \) is an eigenvector of A.
  2. If \( \lambda \) is an eigenvalue of a matrix A, then the linear system \( (\lambda I - A)x = 0 \) has only the trivial solution.
  3. If the characteristic polynomial of a matrix A is \( p(\lambda) = \lambda^2 + 1 \), then A is invertible.
  4. If \( \lambda \) is an eigenvalue of a matrix A, then the eigenspace of A corresponding to \( \lambda \) is the set of eigenvectors of A corresponding to \( \lambda \).
  5. The eigenvalues of a matrix A are the same as the eigenvalues of the reduced row echelon form of A.
  6. If \( 0 \) is an eigenvalue of a matrix A, then the set of columns of A is linearly independent.

Exercise 25 : Working with technology

For the given matrix A, find the characteristic polynomial and the eigenvalues, and find bases for the eigenspaces.

\[ A = \begin{bmatrix} -8 & 33 & 38 & 173 & -30 \\ 0 & 0 & -1 & -4 & 0 \\ 0 & 0 & -5 & -25 & 1 \\ 0 & 0 & 0 & 5 & 0 \\ 4 & -16 & -19 & -86 & 15 \end{bmatrix} \]

Exercise 26 : Working with technology 2

The Cayley–Hamilton Theorem states that every square matrix satisfies its characteristic equation; that is, if A is an \( n \times n \) matrix whose characteristic equation is

\[ \lambda^n + c_1\lambda^{n-1} + \dots + c_n = 0 \]

then

\[ A^n + c_1 A^{n-1} + \dots + c_n = 0. \]

  1. Verify the Cayley–Hamilton Theorem for the matrix

    \[ A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 2 & -5 & 4 \end{bmatrix} \]

  2. Use the result in Exercise 16 to prove the Cayley–Hamilton Theorem for \( 2 \times 2 \) matrices.