Page 263 - Matrix Analysis & Applied Linear Algebra

258 Chapter 4 Vector Spaces

4.8.8. Let λ be a scalar such that (C − λI) is singular.
n×n
(a) If B - C, prove that (B − λI) is also singular.
(b) Prove that (B − λ i I) is singular whenever B n×n is similar to
 0 0 
λ 1 ···
 0 λ 2 ··· 0 
D =  . . . .  .
 . . .
. . . . . 
0 0 ··· λ n
k
k
4.8.9. If A - B, show that A - B for all nonnegative integers k.

4.8.10. Suppose B = {x 1 , x 2 ,..., x n } and B = {y 1 , y 2 ,..., y n } are bases for
m×1
an n -dimensional subspace V⊆ , and let X m×n and Y m×n be

the matrices whose columns are the vectors from B and B , respectively.
T
(a) Explain why Y Y is nonsingular, and prove that the change
T −1 T
of basis matrix from B to B is P = Y Y Y X.

(b) Describe P when m = n.
k
4.8.11. (a) N is nilpotent of index k when N = 0 but N k−1 = 0. If N
n
is a nilpotent operator of index n on , and if N n−1 (y) = 0,
2 n−1 n
show B = y, N(y), N (y),..., N (y) is a basis for ,
and then demonstrate that
00 ··· 00
 
 10
··· 00 


[N] B = J =  01 ··· 00  .
 . . . . . . . . . 

. . . . . . 
00 ··· 10
(b) If A and B are any two n × n nilpotent matrices of index n,
explain why A - B.
(c) Explain why all n × n nilpotent matrices of index n must have
a zero trace and be of rank n − 1.
2
n
4.8.12. E is idempotent when E = E. For an idempotent operator E on ,
r n−r
let X = {x i } and Y = {y i } be bases for R (E) and N (E),
i=1 i=1
respectively.
n
(a) Prove that B = X∪Y is a basis for . Hint: Show Ex i = x i
and use this to deduce that B is linearly independent.
0
(b) Show that [E] B = I r .
0 0
(c) Explain why two n × n idempotent matrices of the same rank
must be similar.
(d) If F is an idempotent matrix, prove that rank (F)= trace (F).