.. _eigHW:

Eigenvalue and Eigenvector Homework
=====================================


#. Consider the following matrix.

   .. math:: \mathbf{A} = \begin{bmatrix}
                             9  & 0 \\ 
                             7  & 5
                          \end{bmatrix}

   ::

        A = np.array([[9, 0], [7, 5]]).astype(float)

   a. Find its eigenvalues using equation :eq:`eq-eigval2x2`.
      Use Python to find its eigenvalues and eigenvectors.

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

   c. Use Python to verify that :math:`\mathbf{A}\, \bm{x} = \lambda \bm{x}` for
      each eigenvalue, eigenvector pair.

#. Use Python's ``roots`` function to find the roots to the following
   polynomial.

   .. math:: x^4 + 4\,x^3 - 7\,x^2 - 22\,x + 24 = 0

#. 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?

   a. 

      .. math::

         \mathbf{A} = \begin{bmatrix}
                         -16  & -3 & 49 \\ 
                         5  & 2 & -15 \\ 
                         -5  & -1 & 16
                      \end{bmatrix}

      ::

        A = np.array([[-16, -3, 49], [5, 2, -15], [-5, -1, 16]]).astype(float)

   b. 

      .. math::

         \mathbf{S} = \begin{bmatrix}
                         24  & 4 & -4 \\ 
                         4  & -9 & 8 \\ 
                         -4  & 8 & 18
                      \end{bmatrix}

      ::

        S = np.array([[24, 4, -4], [4, -9, 8], [-4, 8, 18]]).astype(float)

#. Making use of the diagonalization of the following matrix, find
   :math:`\mathbf{A}^{20}`. You will find that the eigenvalues and
   eigenvectors are complex.

   .. math::

      \mathbf{A} = \begin{bmatrix}
                      -3  & 4 & 2 \\ 
                      -2  & 6 & 3 \\ 
                      2  & -5 & 2
                   \end{bmatrix}

   ::

        A = np.array([[-3, 4, 2], [-2, 6, 3], [2, -5, 2]]).astype(float)

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

   .. math::

      \mathbf{A} = \begin{bmatrix}
                      1.25  & 0.5 \\ 
                      -0.25  & 0.5
                   \end{bmatrix} \qquad
      \bm{u}_0 = \begin{bmatrix}
                   2 \\ 1 
                 \end{bmatrix}

   ::

        A = np.array([[1.25, 0.5], [-0.25, 0.5]]).astype(float)

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

   a. Using the change of basis method, find a general equation for the
      difference equation, :math:`\bm{u}_k`.

   b. What is the steady state of the system?