Linear Algebra, Theory And Applications, 2012a

Numerical Methods For Finding
Eigenvalues
15.1 The Power Method For Eigenvalues
This chapter discusses numerical methods for finding eigenvalues. However, to do this
correctly, you must include numerical analysis considerations which are distinct from linear
algebra. The purpose of this chapter is to give an introduction to some numerical methods
without leaving the context of linear algebra. In addition, some examples are given which
make use of computer algebra systems. For a more thorough discussion, you should see
books 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 }
. ..
J m
and that |λ 1 | > |λ k | for all k. Also let the Jordan form of A be
⎛
⎜
J = ⎝
J 1
⎞
⎟
⎠
with
where N r k
k ≠ 0 but N r k+1
k
=0. Alsolet
J k = λ k I k + N k
P −1 AP = J, A = PJP −1 .
Now fix x ∈ F p . Take Ax and let s 1 be the entry of the vector Ax which has largest
absolute value. Thus Ax/s 1 is a vector y 1 which has a component of 1 and every other
entry of this vector has magnitude no larger than 1. If the scalars {s 1 , ··· ,s n−1 } and
vectors {y 1 , ··· , y n−1 } have been obtained, let
y n ≡ Ay n−1
s n
where s n is the entry of Ay n−1 which has largest absolute value. Thus
y n = AAy n−2 A n x
···=
(15.1)
s n s n−1 s n s n−1 ···s 1
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