Page 405 - Matrix Analysis & Applied Linear Algebra

5.10 Range-Nullspace Decomposition 401

5.10.5. Find a core-nilpotent decomposition and the Drazin inverse of

 
−20 −4
A =  42 4  .
32 2

5.10.6. Fora square matrix A, any scalar λ that makes A − λI singular
is called an eigenvalue for A. The index of an eigenvalue λ is de-
fined to be the index of the associated matrix A − λI. In other words,
index(λ)= index(A − λI). Determine the eigenvalues and the index of
each eigenvalue for the following matrices:

   
10000 11000
 01000   01100 
(a) J =  00100  . (b) J =  00100  .




   
00020 00021
00002 00002
5.10.7. Let P bea projector different from the identity.
(a) Explain why index(P)=1. What is the index of I?
(b) Determine the core-nilpotent decomposition for P.

5.10.8. Let N bea nilpotent matrix of index k, and suppose that x is a vector
such that N k−1 x = 0. Prove that the set
2
C = {x, Nx, N x, ..., N k−1 x}

is a linearly independent set. C is sometimes called a Jordan chain or
a Krylov sequence.

k
5.10.9. Let A be a square matrix of index k, and let b ∈ R A .
(a) Explain why the linear system Ax = b must be consistent.

k
D
(b) Explain why x = A b is the unique solution in R A .
D
(c) Explain why the general solution is given by A b + N (A).
5.10.10. Suppose that A is a square matrix of index k, and let A D be the
Drazin inverse of A as defined in Example 5.10.5. Explain why AA D
k k D
is the projector onto R A along N A . What does I − AA
project onto and along?