Linear Algebra I: Homework 9

Due: Friday, October 27, 2017
  1. Find the eigenvalues and their corresponding eigenspaces for the matrix,

    The characteristic polynomial of is , so the eigenvalues are .

    The eigenspace for eigenvalue 2 is two-dimensional and is spanned by the vectors found by solving the equation ; it is,

    The eigenspace for eigenvalue 3 is one-dimensional and is spanned by the solution to the equation ; it is,

  2. Remember that the eigenvalues of the matrix,

    are , with eigenvector for ,

    and eigenvectors for ,

    Find an invertible matrix and a diagonal matrix so that .

    Based on the known eigenvalues, one choice for is,

    Based on this choice of , one valid choice for is,

  3. Can the matrix

    be diagonalized? Explain why or why not.

    The characteristic polynomial of is, , so the eigenvalues are . However, the only eigenvector of is

    As there is only one eigenvector for the matrix , it is not diagonalizable.

  4. Can the matrix

    be diagonalized? Explain why or why not.

    The characteristic polynomial of is, . This polynomial has discriminant “” of . As the discriminant is greater than zero there are two unique real roots to the polynomial, and hence two different real eigenvalues for . As each eigenvalue has at least one eigenvector, has at least two eigenvectors. So, as is a matrix, this means that yes, is diagonalizable.

  5. For a positive integer and the matrix,

    find a formula for .

    diagonalizes as the matrix product,

    Since , a formula for is,