eigenvalue (of a matrix)

eigenvalue (of a matrix)

Definition

Let

A

be an

complex

n

×

n

matrix. A number

λ

∈

ℂ

is said to be an

eigenvalue

of

A

if there is a nonzero

n

×

1

column vector

x

for which

A

⁢

x

=

λ

⁢

x

.

Computation of Eigenvalues

The computation of the eigenvalues of a given matrix

A

is relatively easy from a theoretical point of view, though often computationally infeasible, or at least difficult. The basic procedure is to note that the eigenvalues of a matrix are precisely the solutions to the equation

det

⁡

(

λ

⁢

I

-

A

)

=

0

.

where

I

denotes the

n

×

n

identity matrix

and

det

is the

determinant

function

. As the above determinant is simply a

polynomial

(of degree

n

, called the

characteristic polynomial

of

A

) in

λ

with

coefficients

in

ℂ

, its roots can be calculated or approximated accordingly to give the eigenvalues of the matrix. Following this train of thought, we also note that this polynomial has degree at least 1, so since

ℂ

is

algebraically closed

, it is thus guaranteed that any

A

has at least one eigenvalue (and at most

n

). If

λ

is a

multiple root

(say, of

multiplicity

k

) of the defining polynomial, we say that

λ

is an

eigenvalue of multiplicity

k

.

If one is given a

n

×

n

matrix

A

of

real numbers

, the above

argument

implies that

A

has at least one complex eigenvalue; the question of whether or not

A

has real eigenvalues is more subtle since there is no real-numbers analogue of the

fundamental theorem of algebra

. It should not be a surprise then that some

real matrices

do not have real eigenvalues. For example, let

A

=

(

0

-

1

1

0

)

.

In this case

det

⁡

(

λ

⁢

I

-

A

)

=

λ

2

+

1

; clearly no real number

λ

λ

2

+

1

=

0

; hence

A

has no real eigenvalues (although

A

has complex eigenvalues

i

and

-

i

).

If one converts the above into an algorithm for calculating the eigenvalues of a matrix

A

, one is led to a two-step procedure:

•

Compute the polynomial

det

⁡

(

λ

⁢

I

-

A

)

.

•

Solve

det

⁡

(

λ

⁢

I

-

A

)

=

0

.

Unfortunately, computing

n

×

n

determinants and finding roots of polynomials of degree

n

are both computationally messy procedures for even moderately large

n

, so for most practical purposes

variations

on this naive are needed. See the

eigenvalue problem

for more .

Properties

•

A

and

A

T

have the same eigenvalues.

•

Eigenvalues of Hermitian matrices are real

.

•

Eigenvalues of

skew-symmetric

are

purely imaginary

(or zero).

•

A real matrix

A

is

diagonalizable

if all of

A

’s eigenvalues are real and distinct.

•

If a

symmetric matrix

has distinct eigenvalues, it is diagonalizable.

Title

eigenvalue (of a matrix)

Canonical name

EigenvalueofAMatrix

Date of creation

2013-03-22 13:42:57

Last modified on

2013-03-22 13:42:57

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

11

Author

mathcam (2727)

Entry type

Definition

Classification

msc 65-00

Classification

msc 15A18

Classification

msc 15-00

Classification

msc 65F15

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