Page 144 - 35Linear Algebra

144 Matrices

to define
0
M = I ,
0
the identity matrix, just like x = 1 for numbers.
As a result, any polynomial can be have square matrices in it’s domain.

3
2
Example 90 Let f(x) = x − 2x + 3x and

1 t
M = .
0 1
Then

1 2t 3 1 3t
2
M = , M = , . . .
0 1 0 1
and so

1 t 1 2t 1 3t
f(M) = − 2 + 3
0 1 0 1 0 1

2 6t
= .
0 2
Suppose f(x) is any function defined by a convergent Taylor Series:

1
00
0
2
f(x) = f(0) + f (0)x + f (0)x + · · · .
2!
Then we can define the matrix function by just plugging in M:
1
00
0
2
f(M) = f(0) + f (0)M + f (0)M + · · · .
2!
There are additional techniques to determine the convergence of Taylor Series
of matrices, based on the fact that the convergence problem is simple for
diagonal matrices. It also turns out that the matrix exponential

1 1
3
2
exp(M) = I + M + M + M + · · · ,
2 3!
always converges.

Matrix Exponential Example

144