5Eigenvalues and eigenvectors

IA Vectors and Matrices

5.1 Preliminaries and definitions

Theorem

(Fundamental theorem of algebra)

.

Let

p

(

z

) be a polynomial of degree

m ≥ 1, i.e.

p(z) =

m

X

j=0

c

j

z

j

,

where c

j

∈ C and c

m

6= 0.

Then

p

(

z

) = 0 has precisely

m

(not necessarily distinct) roots in the complex

plane, accounting for multiplicity.

Note that we have the disclaimer “accounting for multiplicity”. For example,

x

2

−

2

x

+ 1 = 0 has only one distinct root, 1, but we say that this root has

multiplicity 2, and is thus counted twice. Formally, multiplicity is defined as

follows:

Definition

(Multiplicity of root)

.

The root

z

=

ω

has multiplicity

k

if (

z −ω

)

k

is a factor of p(z) but (z − ω)

k+1

is not.

Example.

Let

p

(

z

) =

z

3

− z

2

− z

+ 1 = (

z −

1)

2

(

z

+ 1). So

p

(

z

) = 0 has roots

1, 1, −1, where z = 1 has multiplicity 2.

Definition

(Eigenvector and eigenvalue)

.

Let

α

:

C

n

→ C

n

be a linear map

with associated matrix A. Then x 6= 0 is an eigenvector of A if

Ax = λx

for some

λ

.

λ

is the associated eigenvalue. This means that the direction of the

eigenvector is preserved by the mapping, but is scaled up by λ.

There is a rather easy way of finding eigenvalues:

Theorem. λ is an eigenvalue of A iff

det(A − λI) = 0.

Proof.

(

⇒

) Suppose that

λ

is an eigenvalue and

x

is the associated eigenvector.

We can rearrange the equation in the definition above to

(A − λI)x = 0

and thus

x ∈ ker(A − λI)

But

x 6

=

0

. So

ker

(

A−λI

) is non-trivial and

det

(

A−λI

) = 0. The (

⇐

) direction

is similar.

Definition

(Characteristic equation of matrix)

.

The characteristic equation of

A is

det(A − λI) = 0.

Definition

(Characteristic polynomial of matrix)

.

The characteristic polynomial

of A is

p

A

(λ) = det(A − λI).

From the definition of the determinant,

p

A

(λ) = det(A − λI)

= ε

j

1

j

2

···j

n

(A

j

1

1

− λδ

j

1

1

) ···(A

j

n

n

− λδ

j

n

n

)

= c

0

+ c

1

λ + ··· + c

n

λ

n

for some constants c

0

, ··· , c

n

. From this, we see that

(i) p

A

(

λ

) has degree

n

and has

n

roots. So an

n ×n

matrix has

n

eigenvalues

(accounting for multiplicity).

(ii)

If

A

is real, then all

c

i

∈ R

. So eigenvalues are either real or come in

complex conjugate pairs.

(iii) c

n

= (

−

1)

n

and

c

n−1

= (

−

1)

n−1

(

A

11

+

A

22

+

···

+

A

nn

) = (

−

1)

n−1

tr

(

A

).

But c

n−1

is the sum of roots, i.e. c

n−1

= (−1)

n−1

(λ

1

+ λ

2

+ ···λ

n

), so

tr(A) = λ

1

+ λ

2

+ ··· + λ

n

.

Finally,

c

0

=

p

A

(0) =

det

(

A

). Also

c

0

is the product of all roots, i.e.

c

0

= λ

1

λ

2

···λ

n

. So

det A = λ

1

λ

2

···λ

n

.

The kernel of the matrix

A − λI

is the set

{x

:

Ax

=

λx}

. This is a vector

subspace because the kernel of any map is always a subspace.

Definition

(Eigenspace)

.

The eigenspace denoted by

E

λ

is the kernel of the

matrix A − λI, i.e. the set of eigenvectors with eigenvalue λ.

Definition

(Algebraic multiplicity of eigenvalue)

.

The algebraic multiplicity

M

(

λ

) or

M

λ

of an eigenvalue

λ

is the multiplicity of

λ

in

p

A

(

λ

) = 0. By the

fundamental theorem of algebra,

X

λ

M(λ) = n.

If M(λ) > 1, then the eigenvalue is degenerate.

Definition

(Geometric multiplicity of eigenvalue)

.

The geometric multiplicity

m

(

λ

) or

m

λ

of an eigenvalue

λ

is the dimension of the eigenspace, i.e. the

maximum number of linearly independent eigenvectors with eigenvalue λ.

Definition (Defect of eigenvalue). The defect ∆

λ

of eigenvalue λ is

∆

λ

= M(λ) − m(λ).

It can be proven that ∆

λ

≥

0, i.e. the geometric multiplicity is never greater

than the algebraic multiplicity.