Page 411 - Matrix Analysis & Applied Linear Algebra

5.11 Orthogonal Decomposition 407

URV Factorization

m×n
For each A ∈ of rank r, there are orthogonal matrices U m×m
and V n×n and a nonsingular matrix C r×r such that

0 T
T
C r×r
A = URV = U V . (5.11.11)
0 0
m×n
• The first r columns in U are an orthonormal basis for R (A).
T
• The last m−r columns of U are an orthonormal basis for N A .
T
• The first r columns in V are an orthonormal basis for R A .
• The last n − r columns of V are an orthonormal basis for N (A).
Each different collection of orthonormal bases for the four fundamental
subspaces of A produces a different URV factorization of A. In the
complex case, replace ( ) T by ( ) and “orthogonal” by “unitary.”
∗

Example 5.11.2
Problem: Explain how to make C lower triangular in (5.11.11).
Solution: Apply Householder (or Givens) reduction to produce an orthogonal

B
matrix P m×m such that PA = , where B is r × n of rank r. House-
0
holder (or Givens) reduction applied to B T results in an orthogonal matrix
Q n×n and a nonsingular upper-triangular matrix T such that
T

T
T
QB = T r×r =⇒ B = T | 0 Q =⇒ B = T 0 Q,
0 0 0 0
B T T 0
so A = P T = P T Q is a URV factorization.
0 0 0
Note: C can in fact be made diagonal—see (p. 412).
Have you noticed the duality that has emerged concerning the use of fun-
n n
damental subspaces of A to decompose (or C )? On one hand there is
the range-nullspace decomposition (p. 394), and on the other is the orthogo-
nal decomposition theorem (p. 405). Each produces a decomposition of A. The
n
range-nullspace decomposition of produces the core-nilpotent decomposition
of A (p. 397), and the orthogonal decomposition theorem produces the URV
factorization. In the next section, the URV factorization specializes to become