Mathematics 21b Review Problems

Fall 97 Exam 2

(1) True or False

If A is a square matrix such that A^10 is the zero matrix, then 0 is an eigenvalue of A.
For any square matrix A, the transpose of A^T A is A A^T.
If a 2 x 2 matrix is not invertible, then its trace is an eigenvalue.
If A is an m x n matrix with A A^T = I_n, then the rows of A are orthogonal unit vectors.
The 2 x 2 matrix P representing orthogonal projection onto a line L in the plane is an orthogonal matrix.

The determinant of the following matrix is 1:

(2) Consider the discrete dynamic system x(t+1)= Ax(t), A=

                  1  -1
                  1  k-1

where k>0 is an arbitary positive real number. For the following values of k, describe the phase portrait of the dynamic system. (1)k=1/2 , (2)k=1 , (3)k=2 , (4)k=8.

(3) Consider the linear system Ax=b with

                1  1  0                 3
           A = -1  0  1             b = 1
                0  1  1                 1
                1  2  1                 1

Find all least square solutions x to this linear system.
Compute ||Ax-b|| for these solutions.

(4) Let A=

                       0 1 0
                      -1 2 0
                       0 1 3

Find the eigenvalues of A and their algebraic multiplicities.
Calculate the geometric multiplicity of each eigenvalue. Is there an eigenbasis for A?

(5) Let A be a 3x5 matrix and B be a 5x3 matrix such that AB = I_3. In the exercises we consider the eigenvalues and eigenvectors of the matrix BA.

Explain why rank(A) = 3 and rank(B) = 3.
Show that the nonzero vectors in ker(A) are eigenvectors of BA. What are the corresponing eigenvalues?
Show that the nonzero vectors in im(B) are eigenvectors of BA. What are the corresponding eigenvalues?
Using your answers to above questions, list all eigenvalues of BA together with their geometric multiplicities. Is there an eigenbasis of BA?