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Copyright c 2000 SIAM
CHAPTER
7
Eigenvalues
and
Eigenvectors
7.1
ELEMENTARY PROPERTIES OF EIGENSYSTEMS
Up to this point, almost everything was either motivated by or evolved from the
consideration of systems of linear
algebraic
equations. But we have come to a
turning point, and from now on the emphasis will be different. Rather than being
concerned with systems of
algebraic
equations, many topics will be motivated
or driven by applications involving systems of linear
differential
equations and
their discrete counterparts, difference equations.
For example, consider the problem of solving the system of two first-order
linear differential equations,
du
1
/dt
= 7
u
1
−
4
u
2
and
du
2
/dt
= 5
u
1
−
2
u
2
.
In
matrix notation, this system is
u
1
u
2
=
7
−
4
5
−
2
u
1
u
2
or, equivalently,
u
=
Au
,
(7
.
1
.
1)
where
u
=
u
1
u
2
,
A
=
7
−
4
5
−
2
,
and
u
=
u
1
u
2
.
Because solutions of a single
equation
u
=
λu
have the form
u
=
α
e
λt
,
we are motivated to seek solutions
of (7.1.1) that also have the form
u
1
=
α
1
e
λt
and
u
2
=
α
2
e
λt
.
(7
.
1
.
2)
Differentiating these two expressions and substituting the results in (7.1.1) yields
α
1
λ
e
λt
= 7
α
1
e
λt
−
4
α
2
e
λt
α
2
λ
e
λt
= 5
α
1
e
λt
−
2
α
2
e
λt
⇒
α
1
λ
= 7
α
1
−
4
α
2
α
2
λ
= 5
α
1
−
2
α
2
⇒
7
−
4
5
−
2
α
1
α
2
=
λ
α
1
α
2
.

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COPYRIGHTED
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Please report violations to [email protected]
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Copyright c 2000 SIAM
490
Chapter 7
Eigenvalues and Eigenvectors
In other words, solutions of (7.1.1) having the form (7.1.2) can be constructed
provided solutions for
λ
and
x
=
α
1
α
2
in the matrix equation
Ax
=
λ
x
can
be found. Clearly,
x
=
0
trivially satisfies
Ax
=
λ
x
,
but
x
=
0
provides no
useful information concerning the solution of (7.1.1). What we really need are
scalars
λ
and
nonzero
vectors
x
that satisfy
Ax
=
λ
x
.
Writing
Ax
=
λ
x
as (
A
−
λ
I
)
x
=
0
shows that the vectors of interest are the
nonzero
vectors in
N
(
A
−
λ
I
)
.
But
N
(
A
−
λ
I
) contains nonzero vectors if and only if
A
−
λ
I
is singular. Therefore, the scalars of interest are precisely the values of
λ
that
make
A
−
λ
I
singular or, equivalently, the
λ
’s for which det(
A
−
λ
I
) = 0
.
These observations motivate the definition of eigenvalues and eigenvectors.
66
Eigenvalues and Eigenvectors
For an
n
×
n
matrix
A
,
scalars
λ
and vectors
x
n
×
1
=
0
satisfying
Ax
=
λ
x
are called
eigenvalues
and
eigenvectors
of
A
,
respectively,
and any such pair, (
λ,
x
)
,
is called an
eigenpair
for
A
.
The set of
distinct
eigenvalues, denoted by
σ
(
A
)
,
is called the
spectrum
of
A
.

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