Page 401 - Matrix Analysis & Applied Linear Algebra

5.10 Range-Nullspace Decomposition 397

n
Anytime can be written as the direct sum of two complementary sub-
spaces such that one of them is an invariant subspace for a given square matrix A
we have a block-triangular representation for A according to formula (4.9.9) on
p. 263. And if both complementary spaces are invariant under A, then (4.9.10)
says that this block-triangular representation is actually block diagonal.
Herein lies the true value of the range-nullspace decomposition (5.10.1) be-
k k
cause it turns out that if k = index(A), then R A and N A are both
k
invariant subspaces under A.R A is invariant under A because
k k+1 k
A R A = R A = R A ,
k
and N A is invariant because
k k k+1
x ∈ A N A =⇒ x = Aw for some w ∈ N A = N A

k
=⇒ A x = A k+1 w = 0 =⇒ x ∈ N A k
k k
=⇒ A N A ⊆ N A .
This brings us to a matrix decomposition that is an important building
block for developments that culminate in the Jordan form on p. 590.

Core-Nilpotent Decomposition
k
If A is an n × n singular matrix of index k such that rank A = r,
then there exists a nonsingular matrix Q such that

0
C r×r
−1
Q AQ = (5.10.5)
0 N
in which C is nonsingular, and N is nilpotent of index k. In other
words, A is similar to a 2 × 2 block-diagonal matrix containing a non-
singular “core” and a nilpotent component. The block-diagonal matrix
in (5.10.5) is called a core-nilpotent decomposition of A.
Note: When A is nonsingular, k =0 and r = n, so N is not present,
and we can set Q = I and C = A (the nonsingular core is everything).
So (5.10.5) says absolutely nothing about nonsingular matrices.

Proof. Let Q = X | Y , where the columns of X n×r and Y n×n−r constitute

k
bases for R A k and N A , respectively. Equation (4.9.10) guarantees that
Q −1 AQ must be block diagonal in form, and thus (5.10.5) is established. To see
that N is nilpotent, let

U
Q −1 = ,
V