**MATH4141/MATH6651/PHYS5070**

**FW02**

**Assignment
3 Due
date: Monday, Oct. 28**

1. If **A** is an n_{ }by n matrix with an eigenvalue s and
corresponding eigenvector **x **show
that the matrix (**A** - q**I**)^{-1} has an eigenvalue (s -
q)^{-1 }with the same corresponding eigenvector **x **provided q is not equal to one of the eigenvalues of **A**.

2. Let **A** be an n_{ }by n matrix with an eigenvalues s_{1},
s_{2}, ... s_{n} and corresponding eigenvectors **v**^{(1)}, **v**^{(2)}, ..., **v**^{(n)}. If **x**
is any vector with the property that **x**^{T}**v**^{(1)} = 1 show that the matrix
**B** = **A _{ }- **s

**v**^{(i)} = (s_{i} - s_{1})**w**^{(i) }+ s_{1}(**x**^{T}**w**^{(i) }) **v**^{(1)}

for i = 2, ..., n.

3. Use the Gerschgorin circle theorem to estimate
the location of the eigenvalues of the following matrices. Draw a rough diagram to illustrate your
results.

(a) _{} (b) _{}

4. Use mathematical induction to prove the formulae for **y**_{k}, 1/p_{k}^{2}
and μ_{k} given in the notes on the symmetric power method.

5. Write a program to implement the symmetric
power method for finding the dominant eigenvalue and corresponding eigenvector
if a symmetric matrix. Apply it to the
matrix given in part (a) of question 3.
Also apply it to the matrix

_{}

Use
several different choices for the starting vector **x**_{0} in each case and compare the number of iterations
required for convergence.

(**N.B**.
If you follow Algorithm 9.2 of Burden and Faires be aware that the
convergence criterion in Step 6 will not work if the dominant eigenvalue has a
negative sign. Replace it by one of your
own design).

6. Modify
the program from question 5 to implement the inverse power method to find
another eigenvalue and eigenvector for the matrices referred to there. Use an IMSL routine to solve the set of
linear equations. Choose one that allows
you to store the LU decomposition and use it without recalculation. Take the shift q = 9 in both cases.