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5.11 Orthogonal Decomposition 409

URV factorization with V = U = U 1 | U 2 ), then R (A)= R (U 1 )= R (V 1 )=

R A T . Conversely, if R (A)= R A T , perping both sides and using equation

(5.11.5) produces N (A)= N A T , so (5.11.8) yields a URV factorization with
U = V.
Example 5.11.3
n×n
∗
∗
A ∈C is called a normal matrix whenever AA = A A. As illustrated
in Figure 5.11.2, normal matrices fill the niche between hermitian and (complex)
RPN matrices in the sense that real-symmetric ⇒ hermitian ⇒ normal ⇒ RPN,
with no implication being reversible—details are called for in Exercise 5.11.13.
RPN
Normal
Hermitian

Real-Symmetric Nonsingular

Figure 5.11.2
Exercises for section 5.11

2 1 1
5.11.1. Verify the orthogonal decomposition theorem for A= −1 −1 0 .
−2 −1 −1

5.11.2. Foran inner-product space V, what is V ? What is 0 ?
⊥
⊥
     
1 2
 
2
4
,
5.11.3. Find a basis for the orthogonal complement of M=span     .
0 1
 
3 6
⊥
5.11.4. Forevery inner-product space V, prove that if M⊆V, then M is
a subspace of V.
5.11.5. If M and N are subspaces of an n-dimensional inner-product space,
prove that the following statements are true.
⊥
(a) M⊆N =⇒N ⊥ ⊆M .
(b) (M + N) = M ∩N .
⊥
⊥
⊥
(c) (M∩N) = M + N .
⊥
⊥
⊥