**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 **x**_{0} = (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
λ_{1} and λ_{2} 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 k^{th}
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 λ_{1}
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 λ_{2}. 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 A^{T}A
show that A**v** is an eigenvector of AA^{T}
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**