similar matrix

similar matrix

Definition

A

square matrix

A

is

similar

(or

conjugate

) to a square matrix

B

if there exists a

nonsingular

square matrix

S

such that

A

=

S

-

1

⁢

B

⁢

S

.

(1)

Note that, given

S

as above, we can define

R

=

S

-

1

and have

A

=

R

⁢

B

⁢

R

-

1

. Thus, whether the

inverse

comes first or last does not matter.

Transformations

of the form

S

-

1

⁢

B

⁢

S

(or

S

⁢

B

⁢

S

-

1

) are called

similarity transformations

.

Discussion

Similarity

is useful for turning recalcitrant matrices into pliant ones. The

canonical

example is that a

diagonalizable matrix

A

is similar to the

diagonal matrix

of its

eigenvalues

Λ

, with the matrix of its

eigenvectors

acting as the similarity transformation. That is,

A

=

T

⁢

Λ

⁢

T

-

1

(2)

=

[

v

1

v

2

…

v

n

]

⁢

[

λ

1

0

…

0

λ

2

…

⋮

⋮

λ

n

]

⁢

[

v

1

v

2

…

v

n

]

-

1

.

(3)

This follows directly from the equation defining eigenvalues and eigenvectors,

A

⁢

T

=

T

⁢

Λ

.

(4)

If

A

is

symmetric

(

http://planetmath.org/SymmetricMatrix

) for example, then through this transformation, we have turned

A

into the

product

of two

orthogonal matrices

and a diagonal matrix. This can be very useful. As an application, see the solution for the normalizing constant of a multidimensional Gaussian integral.

Properties of similar matrices

1.

Similarity is

reflexive

(

http://planetmath.org/Reflexive

): All square matrices

A

are similar to themselves via the similarity transformation

A

=

I

-

1

⁢

A

⁢

I

, where

I

is the

identity matrix

with the same

dimensions

as

A

.

2.

Similarity is symmetric (

http://planetmath.org/Symmetric

): If

A

is similar to

B

, then

B

is similar to

A

, as we can define a matrix

R

=

S

-

1

and have

B

=

R

-

1

⁢

A

⁢

R

(5)

3.

Similarity is

transitive

(

http://planetmath.org/Transitive3

): If

A

is similar to

B

, which is similar to

C

, we have

A

=

S

-

1

⁢

B

⁢

S

=

S

-

1

⁢

(

R

-

1

⁢

C

⁢

R

)

⁢

S

=

(

S

-

1

⁢

R

-

1

)

⁢

C

⁢

(

R

⁢

S

)

=

(

R

⁢

S

)

-

1

⁢

C

⁢

(

R

⁢

S

)

.

(6)

4.

Because of 1, 2 and 3, similarity defines an

equivalence relation

() on square matrices, partitioning (

http://planetmath.org/

Partition

) the space of such matrices into a

disjoint set

of

equivalence classes

.

5.

If

A

is similar to

B

, then their

determinants

are equal; i.e. (

http://planetmath.org/Ie

),

det

⁡

A

=

det

⁡

B

. This is easily verified:

det

⁡

A

=

det

⁡

(

S

-

1

⁢

B

⁢

S

)

=

det

⁡

(

S

-

1

)

⁢

det

⁡

B

⁢

det

⁡

S

=

(

det

⁡

S

)

-

1

⁢

det

⁡

B

⁢

det

⁡

S

=

det

⁡

B

.

(7)

In fact,

similar matrices

have the same

characteristic polynomial

, which implies this result directly, the determinant being the

constant term

of the characteristic polynomial (up to sign).

6.

Similar matrices represent the same

linear transformation

after a

change of basis

.

7.

It can be shown that a matrix

A

and its

transpose

A

T

are always similar.

Title

similar matrix

Canonical name

SimilarMatrix

Date of creation

2013-03-22 12:24:37

Last modified on

2013-03-22 12:24:37

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

19

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 15A03

Synonym

similarity transformation

Synonym

similar matrices

Synonym

conjugate matrices

Related topic

Eigenvalue

Related topic

Eigenvector

Related topic

EigenvalueProblem

Defines

similar

Defines

conjugate

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