Problems

Problem 1.

Consider the matrix

1/√2 -1/√2]

5/2 7/2

L-1/√2 7/2 5/2

5

A = 1/√2

a) Find a Householder transformation H such that B = HAHT is a tridiagonal

matrix. This involves some bookkeeping, so it may help to use the aid of a computer

for calculating the linear algebra in steps that you describe.

b) Use Gershgorin's second theorem in combination with a similarity transformation

to estimate the spectrum of A. Show that all eigenvalues are distinct and describe

the location of the eigenvalues as accurately as you can.

Hint: The matrix

[100]

0 a 0

0 0 1

combined with B for a suitable a > 0 may be helpful.

Ta

=

c) Computer exercise: Approximate the largest eigenvalue of A using power iteration

and approximate the other two eigenvalues with inverse iteration (a shift is needed

for the middle eigenvalue). Draw a random vector 2) as start vector (for instance,

with the components being independent, standard normal distributed). Include

your implementation code and the output of 10 iterations for each approximation.