Chapter 15

Numerical Methods, Eigenvalues15.1 The Power Method For Eigenvalues

This chapter discusses numerical methods for finding eigenvalues. However, to do thiscorrectly, you must include numerical analysis considerations which are distinct from linearalgebra. The purpose of this chapter is to give an introduction to some numerical methodswithout leaving the context of linear algebra. In addition, some examples are given whichmake use of computer algebra systems. For a more thorough discussion, you should seebooks on numerical methods in linear algebra like some listed in the references.

Let A be a complex p× p matrix and suppose that it has distinct eigenvalues

{λ 1, · · · ,λ m}

and that |λ 1|> |λ k| for all k. Also let the Jordan form of A be

J =

J1

. . .

Jm

with J1 an m1×m1 matrix. Jk = λ kIk +Nk where Nrk

k ̸= 0 but Nrk+1k = 0. Also let P−1AP =

J, A = PJP−1. Now fix x ∈ Fp. Take Ax and let s1 be the entry of the vector Ax whichhas largest absolute value. Thus Ax/s1 is a vector y1 which has a component of 1 andevery other entry of this vector has magnitude no larger than 1. If the scalars {s1, · · · ,sn−1}and vectors {y1, · · · ,yn−1} have been obtained, let yn ≡ Ayn−1/sn where sn is the entry ofAyn−1 which has largest absolute value. Thus

yn =AAyn−2

snsn−1· · ·= Anx

snsn−1 · · ·s1= (15.1)

1snsn−1 · · ·s1

P

Jn

1. . .

Jnm

P−1 x=

λn1

snsn−1 · · ·s1P

λ−n1 Jn

1. . .

λ−n1 Jn

m

P−1x (15.2)

Consider one of the blocks in the Jordan form. First consider the kth of these blocks,k > 1. It equals

λ−n1 Jn

k =rk

∑i=0

(ni

)λ−n1 λ

n−ik Ni

k

which clearly converges to 0 as n→ ∞ since |λ 1|> |λ k|. An application of the ratio test orroot test for each term in the sum will show this. When k = 1, this block is

λ−n1 Jn

1 = λ−n1 Jn

k =r1

∑i=0

(ni

)λ−n1 λ

n−i1 Ni

1 =

(nr1

)[λ−r11 Nr1

1 + en

]411