Begin2.DVI
ben green
(Ben Green)
#1
Matrices with Special Properties
The following is some terminology associated with square matrices Aand B.
(1) If AB =−BA, then Aand Bare called anticommutative.
(2) If AB =BA, then Aand Bare called commutative.
(3) If AB =BA, then Aand Bare called noncommutative.
(4) If Ap=
︷p times︸︸ ︷
AA ···A= ̃ 0 for some positive integer p,
then Ais called nilpotent of order p.
(5) If A^2 =A, then Ais called idempotent.
(6) If A^2 =I, then Ais called involutory.
(7) If Ap+1 =A, then Ais called periodic with period p. The smallest
integer pfor which Ap+1 =Ais called the least period p.
(8) If AT=A, then Ais called a symmetric matrix.
(9) If AT=−A, then Ais called a skew-symmetric matrix.
(10) If A−^1 exists, then Ais called a nonsingular matrix.
(11) If A−^1 does not exist, then Ais called a singular matrix.
(12) If ATA=AA T=I, then Ais called an orthogonal matrix and AT=A−^1.
Example 10-6.
The matrix A=
[
0 0
− 1 − 1
]
is periodic with least period 2 because
A^2 =AA =
[
0 0
− 1 − 1
][
0 0
− 1 − 1
]
=
[
0 0
1 1
]
and A^3 =A^2 A=
[
0 0
1 1
][
0 0
− 1 − 1
]
=A
Example 10-7. The matrix A=
[
− 1 − 1
1 1
]
is nilpotent of index 2 because
A^2 =AA =
[
− 1 − 1
1 1
][
− 1 − 1
1 1
]
=
[
0 0
0 0
]
= ̃ 0
Example 10-8. The matrix
B=
[
− 1 − 1
2 2
]
is idempotent because
B^2 =BB =
[
− 1 − 1
2 2
][
− 1 − 1
2 2
]
=
[
− 1 − 1
2 2
]
=B
