11.9. Eigenvalue and Eigenvector Homework

  1. Consider the following matrix.

    \[\begin{split}\mathbf{A} = \begin{bmatrix} 9 & 0 \\ 7 & 5 \end{bmatrix}\end{split}\]
    A = np.array([[9, 0], [7, 5]]).astype(float)

    1. Find its eigenvalues using equation (4.2). Use Python to find its eigenvalues and eigenvectors.

    2. Form the two characteristic equations and use the null_space function to verify the eigenvectors.

    3. Use Python to verify that \(\mathbf{A}\, \bm{x} = \lambda \bm{x}\) for each eigenvalue, eigenvector pair.

  2. Use Python’s roots function to find the roots to the following polynomial.

    \[x^4 + 4\,x^3 - 7\,x^2 - 22\,x + 24 = 0\]

  3. Find the diagonalization factors of the following two matrices. Verify the factorization by multiplying the factors. Notice that the second matrix is symmetric. How is its diagonalization simplified?

    1. \[\begin{split}\mathbf{A} = \begin{bmatrix} -16 & -3 & 49 \\ 5 & 2 & -15 \\ -5 & -1 & 16 \end{bmatrix}\end{split}\]
      A = np.array([[-16, -3, 49], [5, 2, -15], [-5, -1, 16]]).astype(float)
    2. \[\begin{split}\mathbf{S} = \begin{bmatrix} 24 & 4 & -4 \\ 4 & -9 & 8 \\ -4 & 8 & 18 \end{bmatrix}\end{split}\]
      S = np.array([[24, 4, -4], [4, -9, 8], [-4, 8, 18]]).astype(float)

  4. Making use of the diagonalization of the following matrix, find \(\mathbf{A}^{20}\). You will find that the eigenvalues and eigenvectors are complex.

    \[\begin{split}\mathbf{A} = \begin{bmatrix} -3 & 4 & 2 \\ -2 & 6 & 3 \\ 2 & -5 & 2 \end{bmatrix}\end{split}\]
    A = np.array([[-3, 4, 2], [-2, 6, 3], [2, -5, 2]]).astype(float)

  5. A system’s state is defined by a difference equation given by the following matrix and initial state, \(\bm{u}_k = \mathbf{A}^k\,\bm{u}_0\).

    \[\begin{split}\mathbf{A} = \begin{bmatrix} 1.25 & 0.5 \\ -0.25 & 0.5 \end{bmatrix} \qquad \bm{u}_0 = \begin{bmatrix} 2 \\ 1 \end{bmatrix}\end{split}\]
    A = np.array([[1.25, 0.5], [-0.25, 0.5]]).astype(float)

u0 = np.array([[2.0], [1.0]]).astype(float)

    1. Using the change of basis method, find a general equation for the difference equation, \(\bm{u}_k\).

    2. What is the steady state of the system?