264 CHAPTER 12. SPECTRAL THEORY
Note that (SDS−1)2
= SDS−1SDS−1 = SD2S−1
and (SDS−1)3
= SDS−1SDS−1SDS−1 = SD3S−1,
etc. In general, you can see that (SDS−1)n
= SDnS−1
In other words, An = SDnS−1. Therefore,
A50 = SD50S−1 =
0 −1 −10 1 01 0 1
1 0 0
0 1 00 0 2
50 1 1 1
0 1 0−1 −1 0
.
It is easy to raise a diagonal matrix to a power. 1 0 00 1 00 0 2
50
=
1 0 00 1 00 0 250
.
It follows A50 = 0 −1 −10 1 01 0 1
1 0 0
0 1 00 0 250
1 1 1
0 1 0−1 −1 0
=
250 −1+250 00 1 0
1−250 1−250 1
That isn’t too hard. However, this would have been horrendous if you had tried to multiplyA50 by hand.
This technique of diagonalization is also important in solving the differential equationsresulting from vibrations. Sometimes you have systems of differential equation and whenyou diagonalize an appropriate matrix, you “decouple” the equations. This is very nice. Itmakes hard problems trivial.
The above example is entirely typical. If A = SDS−1 then Am = SDmS−1 and it is easyto compute Dm. More generally, you can define functions of the matrix using power seriesin this way.
12.1.7 The Matrix Exponential
When A is diagonalizable, one can easily define what is meant by eA. Here is how. Youknow
S−1AS = D
where D is a diagonal matrix. You also know that if D is of the formλ 1 0
. . .
0 λ n
(12.12)