Page 225 - Matrix Analysis & Applied Linear Algebra

220 Chapter 4 Vector Spaces

4.5.11. Sylvester’s law of nullity, given by James J. Sylvester in 1884, states
that for square matrices A and B,

max {ν(A),ν(B)}≤ ν(AB) ≤ ν(A)+ ν(B),
where ν(') = dim N (') denotes the nullity.
(a) Establish the validity of Sylvester’s law.
(b) Show Sylvester’s law is not valid for rectangular matrices be-
cause ν(A) >ν(AB) is possible. Is ν(B) >ν(AB) possible?

4.5.12. For matrices A m×n and B n×p , prove each of the following statements:

(a) rank (AB)= rank (A) and R (AB)= R (A)if rank (B)= n.
(b) rank (AB)= rank (B) and N (AB)= N (B)if rank (A)= n.

4.5.13. Perform the following calculations using the matrices:

   
12 1
A =  24  and b =  2  .
12.01 1.01

(a) Find rank (A), and solve Ax = b using exact arithmetic.
T T T
(b) Find rank A A , and solve A Ax=A b exactly.
(c) Find rank (A), and solve Ax = b with 3-digit arithmetic.
T
T
T
T
(d) Find A A, A b, and the solution of A Ax = A b with
3-digit arithmetic.
r
4.5.14. Prove that if the entries of F r×r satisfy |f ij | < 1 for each i (i.e.,

j=1
each absolute row sum < 1), then I + F is nonsingular. Hint: Use the
triangle inequality for scalars |α+β|≤|α|+|β| to show N (I + F)= 0.
W X
4.5.15. If A = , where rank (A)= r = rank (W r×r ), show that
Y Z
there are matrices B and C such that

W WC I
A = = W I | C .
BW BWC B

∞
4.5.16. For a convergent sequence {A k } of matrices, let A = lim k→∞ A k .
k=1
(a) Prove that if each A k is singular, then A is singular.
(b) If each A k is nonsingular, must A be nonsingular? Why?