Department of Mathematical Sciences

21–241 Matrix Algebra Fall 2002 Professor A. Frieze and Professor S. Ta’asan

Test No. 3

Name: Section:

problem points scores
1  25
2  25
3  25
4  25
total  100

1. (25 pts) (a) Let λ be an eigenvalue of an invertible matrix A. Show that λ^{−1} is an eigenvalue of A^{−1}.

Ax = λx  4 pts
A^{−1}Ax = λA^{−1}x  4 pts
A^{−1}x = λ^{−1}x  4 pts

(b) Show that if A^{2} is the zero matrix then the only eigenvalue of A is zero.

Ax = λx  3 pts
A^{2}x = λAx  3 pts
= λ^{2}x  2 pts

So

0 = λ^{2}x  2 pts

and then λ = 0, since x ≠ 0 3 pts.

2. (25 points) Let

A = [ \begin{matrix}2 & 3 \\ 4 & 1 \end{matrix} ]

Diagonalise A.

det A − λI = (λ − 5)(λ + 2) 5 pts

Eigenvalues are 5 and -2 – 4 pts. Eigenvectors are: 5: [1 1] (4 pts) and -2: [3 − 4] (4 pts).

P = [ \begin{matrix}1 & 3 \\ 1 & −4 \end{matrix} ] 4 pts

P^{−1}AP is diagonal. 4 pts.

3. (25 points) An n × n matrix is orthonormal if its columns are an orthonormal set. (a) Show that the eigenvalues of an n × n orthonormal matrix A are ±1. (Hint: consider the length of vector Ax when x is an eigenvector of A.)

Ax = λx  3 pts
x^{T} A^{T} Ax = λ^{2}||x||^{2}  3 pts
x^{T} x = λ^{2}||x||^{2}  3 pts

So λ^{2} = 1 since x ≠ 0 3 pts.

(b) Let U, V be n × n orthonormal matrices Show that UV is also orthonormal.

(UV)^{T} UV = V^{T} U^{T} UV  3 pts
= V^{T} V  3 pts
= I  3 pts

This implies that UV is orthonormal 4pts.

4. (25 points) Let y = [ \begin{matrix}2 \\ 6 \end{matrix} ] and u = [ \begin{matrix}7 \\ 1 \end{matrix} ] Write y as the sum of a vector in Span{u} and a vector orthogonal to u.

y = (y − \frac{y · u}{u · u}u) + \frac{y · u}{u · u}u.
= [ \begin{matrix}−4/5 \\ 28/5 \end{matrix} ] + [ \begin{matrix}14/5 \\ 2/5 \end{matrix} ] .

10 pts for correct method and rest for accuracy. Take only a few points off for numerical mistakes.