Page 410 - Matrix Analysis & Applied Linear Algebra

406 Chapter 5 Norms, Inner Products, and Orthogonality

be orthonormal bases for R A T and N (A), respectively. It follows that
m n
T ∪B N(A) are orthonormal bases for
B R(A) ∪B N( A ) and B R( A ) and ,
T
respectively, and hence

and (5.11.8)
U m×m = u 1 | u 2 | ··· | u m V n×n = v 1 | v 2 |· · · | v n
T
are orthogonal matrices. Now consider the product R = U AV, and notice
T
T
that r ij = u Av j . However, u A = 0 for i = r +1,...,m and Av j = 0 for
i
i
j = r +1,...,n, so
 T T 
u Av 1 ··· u Av r 0 ··· 0
1
1
. . . . .
 . . . . . . 
 . . . . 
 
 T T 0 ··· 0 
T
r
r
R = U AV =  u Av 1 ··· u Av r  . (5.11.9)
 
 0 ··· 0 0 ··· 0 
. . . . .
 
. . . . .
 . . . . . 
0 ··· 0 0 ··· 0
In other words, A can be factored as

T
T
A = URV = U C r×r 0 V . (5.11.10)
0 0
Moreover, C is nonsingular because it is r × r and

C0 T
rank (C)= rank = rank U AV = rank (A)= r.
0 0
For lack of a better name, we will refer to (5.11.10) as a URV factorization.
We have just observed that every set of orthonormal bases for the four
fundamental subspaces defines a URV factorization. The situation is also re-
versible in the sense that every URV factorization of A defines an orthonor-
mal basis for each fundamental subspace. Starting with orthogonal matrices

U = U 1 | U 2 and V = V 1 | V 2 together with a nonsingular matrix C r×r
such that (5.11.10) holds, use the fact that right-hand multiplication by a non-
singular matrix does not alter the range (Exercise 4.5.12) to observe
R (A)= R (UR)= R (U 1 C | 0)= R (U 1 C)= R (U 1 ).
T
By (5.11.5) and Example 5.11.1, N A = R (A) ⊥ = R (U 1 ) ⊥ = R (U 2 ).
Similarly, left-hand multiplication by a nonsingular matrix does not change the
nullspace, so the second equation in (5.11.5) along with Example 5.11.1 yields

T CV T T T
⊥
N (A)= N RV = N 1 = N CV 1 = N V 1 = R (V 1 ) = R (V 2 ),
0

⊥
⊥
and R A T = N (A) = R (V 2 ) = R (V 1 ). A summary is given below.