Math 104, Summer 2010
Homework 2
1. (Trefethen–Bau 1.3). Suppose a matrix R with entries rij is upper-triangular (i.e., rij = 0
for all i > j). Suppose further that R is m × m and invertible. By using equation 1.8 on the
top of p. 8 of the text and considering the span of the first j columns of R, show that R−1
is also upper-triangular. The same result is also true if we replace “upper-triangular” by
“lower-triangular” (rij = 0 for i < j) but you only need to prove it in the upper-triangular
case.
2. Let u, v be vectors in Rn such that ||u|| = 3, ||v|| = 4. What are the smallest and largest
values of u · v?
3. (a) Is the matrix

1
1
−1
A= 
2 1
−1

1 −1 1
1 1
1

1 1 −1
1 −1 −1
unitary?
 
1
0

(b) Write b = 
0 as a linear combination of the columns of A.
0
(c) Find the orthogonal projection of b onto span{a1 , a2 , a3 }, where a1 , ..., a4 denote the
columns of A.
4. (Trefethen–Bau 2.1) Show that if a matrix A ∈ Cm×m is triangular (i.e., upper-triangular
or lower-triangular, as defined in problem 1) and unitary, then it is diagonal (i.e., its entries
aij are 0 for all i 6= j).
5. (Trefethen–Bau 2.3) We say a matrix A ∈ Cm×m is hermitian if A∗ = A. Recall that
λ ∈ C is an eigenvalue of A if Ax = λx for some x 6= 0.
(a) Prove that if A is hermitian, then the eigenvalues of A must be real. (Hint: consider
x∗ Ax.)
(b) Prove that if A is hermitian and x, y are eigenvectors corresponding to distinct eigenvalues of A, then x and y are orthogonal. (Hints: consider x∗ Ay; you may also want to use
part (a).)
6. (Trefethen–Bau 2.4) What can you say about the eigenvalues of a unitary matrix?
7. Prove that (AT )−1 = (A−1 )T .
1