AIMSLectureNotes 2006
Peter J. Olver
8. Numerical Computation of Eigenvalues
In this part, we discuss some practical methods for computing eigenvalues and eigen
vectors of matrices.
Needless to say, we completely avoid trying to solve (or even write
down) the characteristic polynomial equation. The very basic power method and its vari
ants, which is based on linear iteration, is used to effectively approximate selected eigenval
ues. To determine the complete system of eigenvalues and eigenvectors, the remarkable
QR
algorithm, which relies on the Gram–Schmidt orthogonalization procedure, is the method
of choice, and we shall close with a new proof of its convergence.
8.1. The Power Method.
We have already noted the role played by the eigenvalues and eigenvectors in the
solution to linear iterative systems.
Now we are going to turn the tables, and use the
iterative system as a mechanism for approximating the eigenvalues, or, more correctly,
selected eigenvalues of the coefficient matrix. The simplest of the resulting computational
procedures is known as the
power method
.
We assume, for simplicity, that
A
is a complete
†
n
×
n
matrix. Let
v
1
, . . .,
v
n
denote
its eigenvector basis, and
λ
1
, . . . , λ
n
the corresponding eigenvalues. As we have learned,
the solution to the linear iterative system
v
(
k
+1)
=
A
v
(
k
)
,
v
(0)
=
v
,
(8
.
1)
is obtained by multiplying the initial vector
v
by the successive powers of the coefficient
matrix:
v
(
k
)
=
A
k
v
. If we write the initial vector in terms of the eigenvector basis
v
=
c
1
v
1
+
· · ·
+
c
n
v
n
,
(8
.
2)
then the solution takes the explicit form given in Theorem 7.2, namely
v
(
k
)
=
A
k
v
=
c
1
λ
k
1
v
1
+
· · ·
+
c
n
λ
k
n
v
n
.
(8
.
3)
†
This is not a very severe restriction. Most matrices are complete. Moreover, perturbations
caused by round off and/or numerical inaccuracies will almost inevitably make an incomplete
matrix complete.
9/27/07
131
c
circlecopyrt
2006
Peter J. Olver
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Suppose further that
A
has a single dominant
real
eigenvalue,
λ
1
, that is larger than
all others in magnitude, so

λ
1

>

λ
j

for all
j >
1
.
(8
.
4)
As its name implies, this eigenvalue will eventually dominate the iteration (8.3). Indeed,
since

λ
1

k
≫ 
λ
j

k
for all
j >
1
and all
k
≫
0,
the first term in the iterative formula (8.3) will eventually be much larger than the rest,
and so, provided
c
1
negationslash
= 0,
v
(
k
)
≈
c
1
λ
k
1
v
1
for
k
≫
0
.
Therefore, the solution to the iterative system (8.1) will, almost always, end up being a
multiple of the dominant eigenvector of the coefficient matrix.
To compute the corresponding eigenvalue, we note that the
i
th
entry of the iterate
v
(
k
)
is approximated by
v
(
k
)
i
≈
c
1
λ
k
1
v
1
,i
, where
v
1
,i
is the
i
th
entry of the eigenvector
v
1
.
Thus, as long as
v
1
,i
negationslash
= 0, we can recover the dominant eigenvalue by taking a ratio between
selected components of successive iterates:
λ
1
≈
v
(
k
)
i
v
(
k

1)
i
,
provided that
v
(
k

1)
i
negationslash
= 0
.
(8
.
5)
Example 8.1.
Consider the matrix
A
=
−
1
2
2
−
1
−
4
−
2
−
3
9
7
. As you can check, its
eigenvalues and eigenvectors are
λ
1
= 3
,
v
1
=
1
−
1
3
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 Fall '09
 Olver
 Eigenvectors, Vectors, Matrices, Diagonal matrix, Peter J. Olver

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