Revision examples

Comp.145  March 1997  Dr. A.J. Mestel

1. What can be said, at a glance, about the eigenvalues and eigenvectors of the matrix

A = ( \begin{matrix}−2 & 1 & 1 \\ 1 & −2 & 1 \\ 1 & 1 & −2 \end{matrix} ) ?

Find these eigenvalues and eigenvectors explicitly.

Identify the action of A viewed as a transformation matrix of 3−vectors.

2. Show that the equations x = e^{−x} and x = − ln x are identical for x > 0, and using a rough graph illustrate that there is a unique positive solution. Discuss whether you would expect the schemes

x_{n+1} = e^{−x_{n}}  and x_{n+1} = − ln x_{n}  with x_{0} = .5

to converge or not.

Write down a Newton scheme for this problem, and hence find the solution to four decimal places.

3. Find all complex sequences {z_{n}} for n = 1, 2, … satisfying the recurrence relation

z_{n+2} − 6z_{n+1} + 25z_{n} = 0 .

Deduce that such a sequence satisfying, in addition,

z_{0} = 1 and z_{1} = 3

is

z_{n} = 5^{n} cos nα where sin α = \frac{4}{5}, cos α = \frac{3}{5} .

4. Find the inverse of the matrix

( \begin{matrix}1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} ) .