Math 571 (Winter 2009): Homework 1
Due Date: See web page
[i] Find the eigenvalues and eigenvectors of the 2 ×2 matrix
cos(θ)−sin(θ)
sin(θ) cos(θ)
by hand. Verify that the eigenvectors (which will be complex) are orthogonal to each other.
[ii] Repeat the same exercise for the following matrix:
cos(θ) sin(θ)
sin(θ)−cos(θ)
[iii] Suppose Ais given by the following:
A=0a
0 0.
What is the rank of A? Show that A2= 0.
Find a 3 ×3 matrix of rank 2 such that A3= 0.
[iv] Let
A11 A12
A21 A22
be a matrix in block form with A11 and A22 being square matrices of dimension kand m−k,
respectively. What are the dimensions of A12 and A21? For your information, the inverse is
given by the following block matrix:
(A11 −A12A−1
22 A21)−1A−1
11 A12(A21A−1
11 A12 −A22)−1
(A21A−1
11 A12 −A22)−1A21A−1
11 (A22 −A21A−1
11 A12)−1
[v] One can find the inverse of a 4 ×4 matrix by breaking it up into 2 ×2 blocks, using the
formula for the inverse of a 2 ×2 matrix, and applying the result in [iv]. Give an example
of a 4 ×4 matrix which is invertible and full rank, but for which the method fails.
[vi] Browse the web page http://math-atlas.sourceforge.net/ and a write a half-page
report on ATLAS.
Next go to the install folder and read Section 2.3. Try to follow all the steps for in-
stalling the software. Note: The ../configure line must be replaced by ../ATLAS3.8.0/configure.
You may need to play around a little with the options for configure. I only used -b 64
and --prefix. The build took about 2 hours on my system.
If you have a Linux system, the installation is not hard. On a Windows system, you need
to know about cygwin and have a certain degree of skill. On Vista, the install cannot be
completed. On Macs, I don’t know.
You are not expected to complete the install part of the exercise. Give it a try and write
a brief report about how far you got.