Page 387 - 35Linear Algebra

G.6 Matrices 387

so this pair of matrices do commute. Lets try A and C:
2
1 + a a 1 a
AC = , and CA = 2
a 1 a 1 + a
so
AC 6= CA
and this pair of matrices does not commute. Generally, matrices usually do not
commute, and the problem of finding those that do is a very interesting one.

Matrix Exponential Example

This video shows you how to compute

0 θ
exp .
−θ 0
For this we need to remember that the matrix exponential is defined by its
power series
1 2 1 3
exp M := I + M + M + M + · · · .
2! 3!
Now lets call

0 θ
= iθ
−θ 0
where the matrix

0 1
i :=
−1 0
and by matrix multiplication is seen to obey
2
4
3
i = −I , i = −i , i = I .
Using these facts we compute by organizing terms according to whether they
have an i or not:
1 2 1
exp iθ = I + θ (−I) + (+I) + · · ·
2! 4!
1 3 1
+ iθ + θ (−i) + i + · · ·
3! 5!
1 1
4
2
= I(1 − θ + θ + · · · )
2! 4!
1 1
5
3
+ i(θ − θ + θ + · · · )
3! 5!
= I cos θ + i sin θ

cos θ sin θ
= .
− sin θ cos θ
Here we used the familiar Taylor series for the cosine and sine functions. A
fun thing to think about is how the above matrix acts on vector in the plane.
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