Page 27 - A Course in Linear Algebra with Applications

1.2: Operations with Matrices 11

(iv) Powers of a matrix
Once matrix products have been defined, it is clear how to
define a non-negative power of a square matrix. Let A be an
n x n matrix; then the mth power of A, where m is a non-
negative integer, is defined by the equations

m
A 0 = I n and A m+1 = A A.
This is an example of a recursive definition: the first equation
0
specifies A , while the second shows how to define A m+1 , un-
m
der the assumption that A has already been defined. Thus
2
A 1 = A, A 2 = AA, A 3 = A A etc. We do not attempt to
define negative powers at this juncture.
Example 1.2.5
Let

Then

The reader can verify that higher powers of A do not lead
to new matrices in this example. Therefore A has just four
3
distinct powers, A 0 = I 2, A 1 = A, A 2 and A .
(v) The transpose of a matrix
If A is an m x n matrix, the transpose of A,

is the n x m matrix whose (i,j) entry equals the (j,i) entry
T
of A. Thus the columns of A become the rows of A . For
example, if
/a b\
A = c d ,
V fJ