MATH4141/MATH6651/PHYS5070

FW00

Term test 2                                                                 Monday, Nov. 13, 14:30 - 15:30

Calculators may be used as aids.

Note the formula sheet at the end of this test paper.

The total number of marks on this paper is 35.

1. (4 marks)

a) Define a similarity transformation of a square matrix A.

b) What property do similarity transformations have that make them useful in the calculation of eigenvalues?

c) Show that if the square matrix A is factored as A = QR where Q is an orthogonal matrix and R is an upper triangular matrix, then RQ is a similarity transformation of A.

2. (4 marks)  Carry out one full iteration of the symmetric power method to find the eigenvalue of largest magnitude of the matrix:

Take x0 = (1, 0, 0).

3. (6 marks)  Find a rotation matrix P such that the (2, 1) element of PA is zero if A is the matrix

in question 2.

4. (4 marks)  A positive-definite, symmetrix matrix has eigenvalues which are all positive.  Use the Gerschgorin Circle Theorem to derive a sufficient condition for a symmetric matrix to be positive definite.

5. (8 marks)  If λ1 and λ2 are the the largest and second-largest eigenvalues in magnitude of a symmetric matrix A then it can be shown that the symmetric power method yields an approximate eigenvalue

at the kth iteration.  In one of the questions in Assignment 3, the  eigenvalues of the matrix  were  9.34, 3.25, -4.65 and -12.95.

a) Using the above formula, estimate approximately how many iterations would be required to find λ1 to a relative accuracy of 6 significant figures by the symmetric power method.

b) By using the inverse power method with a shift equal to 9 you were able to find λ2.  Estimate how many iterations would be required to find this eigenvalue to a relative accuracy of 6 significant figures.

6. (4 marks)  If v is an eigenvector of ATA show that Av is an eigenvector of AAT with the same eigenvalue.

7. (5 marks)

a) If we apply the Householder method to a symmetric matrix, what is the form of the resulting matrix?

b) What is another method other than the power method or the Householder plus QR algorithm for finding eignevalues of a matrix.

c) Briefly describe how shifts are used in the QR algorithm.

d) Give one possible use of the singular value decomposition of a matrix.

THE END