Page 342 - Matrix Analysis & Applied Linear Algebra

338 Chapter 5 Norms, Inner Products, and Orthogonality

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5.6.13. Let x =(1/3) −2 .
−2
(a) Determine an elementary reflector R such that Rx lies on the
x-axis.
(b) Verify by direct computation that your reflector R is symmet-
ric, orthogonal, and involutory.
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(c) Extend x to an orthonormal basis for by using an elemen-
tary reflector.

5.6.14. Let R = I − 2uu , where u n×1 =1. If x is a fixed point for R in
∗
the sense that Rx = x, and if n> 1, prove that x must be orthogonal
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to u, and then sketch a picture of this situation in .

n×1
5.6.15. Let x, y ∈ be vectors such that x = y but x = y. Explain
how to construct an elementary reflector R such that Rx = y.
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Hint: The vector u that defines R can be determined visually in
by considering Figure 5.6.2.

5.6.16. Let x n×1 be avector such that x =1, and partition x as

x 1
x = , where ˜ x is n − 1 × 1.
˜ x
(a) If the entries of x are real, and if x 1 =1, show that

T
˜ x 1
x 1
P = T , where α =
˜ x I − α˜ x˜x 1 − x 1
is an orthogonal matrix.
(b) Suppose that the entries of x are complex. If |x 1 | =1, and if
µ is the number defined in (5.6.10), show that the matrix

2 ∗
x 1 µ ˜x 1
U = , where α =
∗
˜ x µ(I − α˜ x˜x ) 1 −|x 1 |
is unitary. Note: These results provide an easy way to extend
n
a given vector to an orthonormal basis for the entire space
n
or C .