448 CHAPTER 18. LINEAR TRANSFORMATIONS

Now recall that u3 is a unit vector and so the above equals

1√(a2 +b2)

(−ac,−bc,a2 +b2)

Then from the above, A is given by−ac√(a2+b2)

b√a2+b2

a

−bc√(a2+b2)

−a√a2+b2

b√

a2 +b2 0 c

 cosθ −sinθ 0

sinθ cosθ 00 0 1



−ac√(a2+b2)

b√a2+b2

a

−bc√(a2+b2)

−a√a2+b2

b√

a2 +b2 0 c

−1

Of course the matrix is an orthogonal matrix so it is easy to take the inverse by simplytaking the transpose. Then doing the computation and then some simplification yields

=

 a2 +(1−a2

)cosθ ab(1− cosθ)− csinθ ac(1− cosθ)+bsinθ

ab(1− cosθ)+ csinθ b2 +(1−b2

)cosθ bc(1− cosθ)−asinθ

ac(1− cosθ)−bsinθ bc(1− cosθ)+asinθ c2 +(1− c2

)cosθ

 .

(18.13)With this, it is clear how to rotate clockwise about the unit vector (a,b,c) . Just rotate

counter clockwise through an angle of −θ . Thus the matrix for this clockwise rotation isjust

=

 a2 +(1−a2

)cosθ ab(1− cosθ)+ csinθ ac(1− cosθ)−bsinθ

ab(1− cosθ)− csinθ b2 +(1−b2

)cosθ bc(1− cosθ)+asinθ

ac(1− cosθ)+bsinθ bc(1− cosθ)−asinθ c2 +(1− c2

)cosθ

 .

In deriving 18.13 it was assumed that c ̸=±1 but even in this case, it gives the correctanswer. Suppose for example that c = 1 so you are rotating in the counter clockwise di-rection about the positive z axis. Then a,b are both equal to zero and 18.13 reduces to thecorrect matrix for rotation about the positive z axis.

18.6 The Matrix Exponential, Differential Equations ∗

You want to find a matrix valued function Φ(t) such that

Φ′ (t) = AΦ(t) , Φ(0) = I, A is p× p (18.14)

Such a matrix is called a fundamental matrix.What is meant by the above symbols? The idea is that Φ(t) is a matrix whose entries

are differentiable functions of t. The meaning of Φ′ (t) is the matrix whose entries are thederivatives of the entries of Φ(t). For example, abusing notation slightly,(

t t2

sin(t) tan(t)

)′=

(1 2t

cos(t) sec2 (t)

).

What are some properties of this derivative? Does the product rule hold for example?