Page 341 - Matrix Analysis & Applied Linear Algebra

5.6 Unitary and Orthogonal Matrices 337

5.6.8. (a) Explain why the standard inner product is invariant under a uni-
tary transformation. That is, if U is any unitary matrix, and if
u = Ux and v = Uy, then

∗
u v = x y.
∗
n
(b) Given any two vectors x, y ∈ , explain why the angle between
them is invariant under an orthogonal transformation. That is, if
u = Px and v = Py, where P is an orthogonal matrix, then

cos θ u,v = cos θ x,y .

5.6.9. Let U m×r bea matrix with orthonormal columns, and let V k×n be a
r×k
matrix with orthonormal rows. For an arbitrary A ∈C , solve the
following problems using the matrix 2-norm (p. 281) and the Frobenius
matrix norm (p. 279).
(a) Determine the values of U , V , U , and V .
2 2 F F
(b) Show that UAV = A . (Hint: Start with UA . )
2 2 2
(c) Show that UAV = A .
F F
Note: In particular, these properties are valid when U and V are
unitary matrices. Because of parts (b) and (c), the 2-norm and the F -
norm are said to be unitarily invariant norms.

   
−2 1
5.6.10. Let u =  1  and v =  4  .
3 0
−1 −1
(a) Determine the orthogonal projection of u onto span {v} .
(b) Determine the orthogonal projection of v onto span {u} .
(c) Determine the orthogonal projection of u onto v .
⊥
(d) Determine the orthogonal projection of v onto u .
⊥

∗
5.6.11. Consider elementary orthogonal projectors Q = I − uu .
(a) Prove that Q is singular.
(b) Now prove that if Q is n × n, then rank (Q)= n − 1.
Hint: Recall Exercise 4.4.10.

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5.6.12. Forvectors u, x ∈C such that u =1, let p be the orthogonal
projection of x onto span {u} . Explain why p ≤ x with equality
holding if and only if x is a scalar multiple of u.