Page 409 - Matrix Analysis & Applied Linear Algebra

5.11 Orthogonal Decomposition 405

We are now in a position to understand why the four fundamental subspaces
m×n
associated with a matrix A ∈ are indeed “fundamental.” First consider
⊥
n
R (A) , and observe that for all y ∈ ,
T
T
x ∈ R (A) ⊥ ⇐⇒ Ay x =0 ⇐⇒ y A x =0
7 8
T T
⇐⇒ y A x =0 ⇐⇒ A x = 0 (Exercise 5.3.2)
T
⇐⇒ x ∈ N A .
T
Therefore, R (A) ⊥ = N A . Perping both sides of this equation and replac-
56 T T ⊥
ing A by A produces R A = N (A) . Combining these observations
produces one of the fundamental theorems of linear algebra.
Orthogonal Decomposition Theorem

m×n
For every A ∈ ,
T T
⊥
⊥
R (A) = N A and N (A) = R A . (5.11.5)
m×n
In light of (5.11.1), this means that every matrix A ∈ produces
m n
an orthogonal decomposition of and in the sense that
m ⊥ T
= R (A) ⊕ R (A) = R (A) ⊕ N A , (5.11.6)
and

n
⊥
= N (A) ⊕ N (A) = N (A) ⊕ R A T . (5.11.7)
Theorems without hypotheses tend to be extreme in the sense that they
either say very little or they reveal a lot. The orthogonal decomposition theorem
has no hypothesis—it holds for all matrices—so, does it really say something
significant? Yes, it does, and here’s part of the reason why.
m n
In addition to telling us how to decompose and in terms of the
four fundamental subspaces of A, the orthogonal decomposition theorem also
tells us how to decompose A itself into more basic components. Suppose that
rank (A)= r, and let

B R(A) = {u 1 , u 2 ,..., u r } and B N( A )
T = {u r+1 , u r+2 ,..., u m }
T
be orthonormal bases for R (A) and N A , respectively, and let
B R( A ) and B N(A) = {v r+1 , v r+2 ,..., v n }
T = {v 1 , v 2 ,..., v r }
56
Here, as well as throughout the rest of this section, ( ) T can be replaced by ( ) ∗ whenever
m×n m×n
is replaced by C .