commuting matrices

commuting matrices

We consider the properties of

commuting matrices

and

linear transformations

over a

vector space

V

.
Two linear transformations

φ

i

:

V

→

V

,

i

=

1

,

2

are said to commute if for every

v

∈

V

,

φ

1

⁢

(

φ

2

⁢

(

v

)

)

=

φ

2

⁢

(

φ

1

⁢

(

v

)

)

.

If

V

has finite

dimension

n

and we fix a basis of

V

then we may represent the linear transformations as

n

×

n

matrices

A

i

and here the condition of commuting linear transformations is

equivalent

to testing if their corresponding matrices commute:

A

1

⁢

A

2

=

A

2

⁢

A

1

.

Simultaneous triangularisation of commuting matrices over any field

can be achieved but may require an

extension

of the field. The reason begins to be apparent from the study of

eigenvalues

.

Remark 1

.

Because the

implication

of commuting matrices is best expressed through

eigenvectors

, we prefer the treatment of linear transformations for the .

Recall a linear transformation

f

:

V

→

V

is said to leave a

subspace

E

≤

V

invariant

if

f

⁢

(

E

)

≤

E

.

Proposition 2

.

If

{

φ

}

i

∈

I

are commuting linear transformations and

E

is
an

eigenspace

of

φ

i

0

for some

i

0

∈

I

, then for all

i

∈

I

,

φ

i

⁢

(

E

)

≤

E

.

Proof.

Let

λ

be the eigenvalue of

φ

i

0

on

E

. Take any

i

∈

I

and

v

∈

E

. Then

φ

i

0

⁢

(

φ

i

⁢

(

v

)

)

=

φ

i

⁢

(

φ

i

0

⁢

(

v

)

)

=

φ

i

⁢

(

λ

⁢

v

)

=

λ

⁢

φ

i

⁢

(

v

)

.

Therefore

φ

i

⁢

(

v

)

∈

E

as

E

is the

λ

eigenspace of

φ

i

0

. In particular,

φ

i

⁢

(

E

)

≤

E

.
∎

We have just shown that commuting linear transformations preserve each other’s eigenspaces. This property does not depend on a finite dimension for

V

or a

finite set

of commuting

transformations

. However, to characterize commuting linear transformations further will require that

V

have finite dimension.

Proposition 3

.

Let

V

be a finite dimensional vector space and let

{

φ

}

i

∈

I

be a family of commuting

diagonalizable

linear transformations from

V

to

V

. Then

φ

i

can be simultaneously diagonalized.

Proof.

If a finite dimensional linear transformation is diagonalizable
over its field then it has all its eigenvalues in the field
(under some basis the matrix is

diagonal

and the eigenvalues
are simply those

elements

on the diagonal.)

If all the eigenvalues of a linear transformation are the same then
the associated

diagonal matrix

is scalar. If all

φ

i

are scalar then they are simultaneously diagonalized.

Now presume that each

φ

i

is
not a scalar transformation. Hence there are at least two distinct eigenspaces. It follows each eigenspace of

φ

i

has dimension
less than that of

V

.

Now we set up an

induction

on the dimension of

V

. When the
dimension of

V

is 1, all linear transformations are scalar.
Now suppose that for all vector spaces of dimension

n

, any commuting
diagonalizable linear transformations can be simultaneously
diagonalized. Then in the case where

dim

⁡

V

=

n

+

1

, either all the
linear transformations are scalar and so simultaneously
diagonalized, or at least one is not scalar in which case its eigenspaces
are proper subspaces. Since the maps commute they respect each others eigenspaces. So we restrict the maps to any eigenspace
and by induction simultaneously

diagonalize

on this subspace.
As the linear transformations are diagonalizable, the sum of the eigenspaces of any

φ

i

is

V

so this process simultaneously diagonalizes each of the

φ

i

.
∎

Of course it is possible to have commuting matrices which are not diagonalizable. At the other extreme are unipotent matrices, that is, matrices with all eigenvalues 1. Aside from the

identity matrix

, unipotent matrices are never diagonal. Yet they often commute. But here the

generalized eigenspaces

substitute for the usual eigenspaces.

It is generally not true that two unipotent matrices commute, even if they share the same eigenspace. For example, the set of unitriangular matrices forms a

nilpotent group

which is

abelian

only for

2

×

2

-matrices.

However, if we consider unipotent matrices of the form

[

I

k

*

0

I

j

]

we find these to correspond to

k

×

j

matrices under

addition

. Thus this large family of unipotent matrices do commute.

Title

commuting matrices

Canonical name

CommutingMatrices

Date of creation

2013-03-22 15:54:12

Last modified on

2013-03-22 15:54:12

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

20

Author

Algeboy (12884)

Entry type

Definition

Classification

msc 15A04

Related topic

SimultaneousTriangularisationOfCommutingMatricesOverAnyField2

Related topic

CommonEigenvectorOfADiagonalElementCrossSection

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