Page 308 - Matrix Analysis & Applied Linear Algebra

304 Chapter 5 Norms, Inner Products, and Orthogonality

5.4.5. With respect to the inner product for matrices given by (5.3.2), verify
that the set
!
1 01 1 1 0 1 1 −1 1 11
B = √ , √ , ,
2 10 2 0 −1 2 1 1 2 −11
2×2
is an orthonormal basis for , and then compute the Fourier expan-
1 1
sion of A = with respect to B.
1 1

2 1
5.4.6. Determine the angle between x = −1 and y = 1 .
1 2

5.4.7. Given an orthonormal basis B for a space V, explain why the Fourier
expansion for x ∈V is uniquely determined by B.

n
5.4.8. Explain why the columns of U n×n are an orthonormal basis for C if
∗
and only if U = U −1 . Such matrices are said to be unitary—their
properties are studied in a later section.
5.4.9. Matrices with the property A A = AA ∗ are said to be normal. No-
∗
tice that hermitian matrices as well as real symmetric matrices are in-
cluded in the class of normal matrices. Prove that if A is normal, then
R (A) ⊥ N (A)—i.e., every vector in R (A)is orthogonal to every vec-
tor in N (A). Hint: Recall equations (4.5.5) and (4.5.6).

5.4.10. Using the trace inner product described in Example 5.3.1, determine the
angle between the following pairs of matrices.

10 11
(a) I = and B = .
01 11

13 2 −2
(b) A = and B = .
24 2 0
n
5.4.11. Why is the definition for cos θ given in (5.4.1) not good for C ? Explain
n
how to define cos θ so that it makes sense in C .

5.4.12. If {u 1 , u 2 ,..., u n } is an orthonormal basis for an inner-product space
V, explain why

x y = x u i u i y
i
holds for every x, y ∈V.