Page 145 - Matrix Analysis & Applied Linear Algebra

3.9 ElementaryMatrices and Equivalence 139

Transposition and Rank
Transposition does not change the rank—i.e., for all m × n matrices,

T
rank (A)= rank A and rank (A)= rank (A ). (3.9.11)
∗

Proof. Let rank (A)= r, and let P and Q be nonsingular matrices such that

I r 0 r×n−r
PAQ = N r = .
0 m−r×r 0 m−r×n−r
T
T
T
Applying the reverse order law for transposition produces Q A P T = N .
r
T
T
Since Q T and P T are nonsingular, it follows that A ∼ N , and therefore
r

T T I r 0 r×m−r
rank A = rank N r = rank = r = rank (A).
0 n−r×r 0 n−r×m−r
¯ ¯ ¯
To prove rank (A)= rank (A ), write N r = N r = PAQ = PAQ, and use the
∗
fact that the conjugate of a nonsingular matrix is again nonsingular (because
¯
¯ −1

K = K −1 ) to conclude that N r ∼ A, and hence rank (A)= rank A . It
T
now follows from rank (A)= rank A that
¯

¯ T
rank (A )= rank A = rank A = rank (A).
∗
Exercises for section 3.9
3.9.1. Suppose that A is an m × n matrix.
(a) If [A|I m ] is row reduced to a matrix [B|P], explain why P
must be a nonsingular matrix such that PA = B.
! !
(b) If A is column reduced to C , explain why Q must be a
I n Q
nonsingular matrix such that AQ = C.
(c) Find a nonsingular matrix P such that PA = E A , where
 
1234
A =  2467  .
1236
(d) Find nonsingular matrices P and Q such that PAQ is in rank
normal form.