Page 403 - Matrix Analysis & Applied Linear Algebra

5.10 Range-Nullspace Decomposition 399
Example 5.10.4

Problem: Explain how each noninvertible linear operator defined on an n-
dimensional vector space V can be decomposed as the “direct sum” of an in-
vertible operator and a nilpotent operator.
Solution: Let T be a linear operator of index k defined on V = R⊕N,
k k
where R = R T and N = N T , and let E = T and F = T be
/R /N
the restriction operators as described in §4.9. Since R and N are invariant
subspaces for T, we know from the discussion of matrix representations on
p. 263 that the right-hand side of the core-nilpotent decomposition in (5.10.5)
must be the matrix representation of T with respect to a basis B R ∪B N , where
B R and B N are respective bases for R and N. Furthermore, the nonsingular
matrix C and the nilpotent matrix N are the matrix representations of E and
F with respect to B R and B N , respectively. Consequently, E is an invertible
operator on R, and F is a nilpotent operator on N. Since V = R⊕N, each
x ∈V can be expressed as x = r + n with r ∈R and n ∈N. This allows
us to formulate the concept of the direct sum of E and F by defining E ⊕ F
to be the linear operator on V such that (E ⊕ F)(x)= E(r)+ F(n) for each
x ∈V. Therefore,

T(x)= T(r + n)= T(r)+ T(n)=(T )(r)+(T )(n)
/R /N
= E(r)+ F(n)=(E ⊕ F)(x) for each x ∈V.

In other words, T = E ⊕ F in which E = T is invertible and F = T is
/R /N
nilpotent.
Example 5.10.5

Drazin Inverse. Inverting the nonsingular core C and neglecting the nilpo-
tent part N in the core-nilpotent decomposition (5.10.5) produces a natural
generalization of matrix inversion. More precisely, if
−1
C 0 −1 D C 0 −1
A = Q Q , then A = Q Q (5.10.6)
0 N 0 0
defines the Drazin inverse of A. Even though the components in a core-
nilpotent decomposition are not uniquely defined by A, it can be proven that
A D is unique and has the following properties.

• A D = A −1 when A is nonsingular (the nilpotent part is not present).

D
k
D
D
• A AA D = A , AA D = A A, A k+1 A D = A , where k = index(A). 55
55
These three properties served as Michael P. Drazin’s original definition in 1968. Initially,