Page 412 - Matrix Analysis & Applied Linear Algebra

408 Chapter 5 Norms, Inner Products, and Orthogonality

the singular value decomposition (p. 412), and in a somewhat parallel manner,
the core-nilpotent decomposition paves the way to the Jordan form (p. 590).
These two parallel tracks constitute the backbone for the theory of modern linear
algebra, so it’s worthwhile to take a moment and reflect on them.
n
The range-nullspace decomposition decomposes with square matrices
while the orthogonal decomposition theorem does it with rectangular matrices.
So does this mean that the range-nullspace decomposition is a special case of,
or somehow weaker than, the orthogonal decomposition theorem? No! Even for
square matrices they are not very comparable because each says something that
the other doesn’t. The core-nilpotent decomposition (and eventually the Jordan
form) is obtained by a similarity transformation, and, as discussed in §§4.8–4.9,
similarity is the primary mechanism for revealing characteristics of A that are
independent of bases or coordinate systems. The URV factorization has little
to say about such things because it is generally not a similarity transforma-
tion. Orthogonal decomposition has the advantage whenever orthogonality is
naturally built into a problem—such as least squares applications. And, as dis-
cussed in §5.7, orthogonal methods often produce numerically stable algorithms
for floating-point computation, whereas similarity transformations are generally
not well suited for numerical computations. The value of similarity is mainly on
the theoretical side of the coin.
So when do we get the best of both worlds—i.e., when is a URV factoriza-
tion also a core-nilpotent decomposition? First, A must be square and, second,
(5.11.11) must be a similarity transformation, so U = V. Surprisingly, this
happens for a rather large class of matrices described below.

Range Perpendicular to Nullspace
For rank (A n×n )= r, the following statements are equivalent:
• R (A) ⊥ N (A), (5.11.12)
T
• R (A)= R A , (5.11.13)
T
• N (A)= N A , (5.11.14)

0
C r×r T
• A = U U (5.11.15)
0 0
in which U is orthogonal and C is nonsingular. Such matrices will
be called RPN matrices, short for“range perpendicular to nullspace.”
Some authors call them range-symmetric or EP matrices. Nonsingular
matrices are trivially RPN because they have a zero nullspace. For com-
∗
plex matrices, replace ( ) T by ( ) and “orthogonal” by “unitary.”

Proof. The fact that (5.11.12) ⇐⇒ (5.11.13) ⇐⇒ (5.11.14) is a direct conse-
quence of (5.11.5). It suffices to prove (5.11.15) ⇐⇒ (5.11.13). If (5.11.15) is a