aarjaneiro/planetmath huggingface dataset:
example of linear involution

example of linear involution

Let

V

be the

vector space

of

m

×

n

complex matrices

. Then the operator

L

:

A

↦

A

H

, which takes a matrix

A

∈

V

into its

Hermitian conjugate

A

H

∈

V

is an

involution

. The projection operators induced by this involution decompose a matrix into a

direct sum of Hermitian and skew-Hermitian matrices

.

Title

example of linear involution

Canonical name

ExampleOfLinearInvolution

Date of creation

2013-03-22 14:14:11

Last modified on

2013-03-22 14:14:11

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Example

Classification

msc 15A21

Related topic

BanachAlgebra

Generated on Fri Feb 9 18:26:36 2018 by

LaTeXML
linear involution

linear involution

Definition.

Let

V

be a

vector space

.
A

linear involution

is a

linear
operator

L

:

V

→

V

such that

L

2

is the

identity operator

on

V

.
An

equivalent

definition is that a linear involution is a linear operator that
equals its own

inverse

.

Theorem

1.

Let

V

be a vector space and let

A

:

V

→

V

be a linear involution.
Then the

eigenvalues

of

A

are

±

1

. Further,
if

V

is

ℂ

n

, and

A

is a

n

×

n

complex matrix

, then we have that:

1.

det

⁡

A

=

±

1

.

2.

The

characteristic polynomial

of

A

,

p

⁢

(

λ

)

=

det

⁡

(

A

-

λ

⁢

I

)

,
is a

reciprocal polynomial

, i.e.,

p

⁢

(

λ

)

=

±

λ

n

⁢

p

⁢

(

1

/

λ

)

.

(proof. (

http://planetmath.org/EigenvaluesOfAnInvolution

))

The next theorem gives a correspondence between

involution

operators

and projection operators.

Theorem 2.

Let

L

and

P

be linear operators on a
vector space

V

over a field of

characteristic

not 2, and let

I

be the identity operator on

V

.
If

L

is an involution then
the operators

1

2

⁢

(

I

±

L

)

are projection operators.

Conversely

, if

P

is a projection operator, then
the operators

±

(

2

⁢

P

-

I

)

are involutions.

Involutions have important application in expressing hermitian-orthogonal operators, that is,

H

t

=

H

¯

=

H

-

1

. In fact, it may be represented as

H

=

L

⁢

e

i

⁢

S

,

being

L

a real

symmetric

involution operator and

S

a real

skew-symmetric

operator

permutable

with it, i.e.

L

=

L

¯

=

L

t

,

L

2

=

I

,

S

=

S

¯

=

-

S

t

,

L

⁢

S

=

S

⁢

L

.

Title

linear involution

Canonical name

LinearInvolution

Date of creation

2013-03-22 13:34:37

Last modified on

2013-03-22 13:34:37

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

14

Author

matte (1858)

Entry type

Definition

Classification

msc 15A21

Synonym

involution

Related topic

Projection

Related topic

AntiIdempotent

Generated on Fri Feb 9 18:26:38 2018 by

LaTeXML
normal matrix

normal matrix

A

complex matrix

A

∈

ℂ

n

×

n

is said to be

normal

if

A

∗

⁢

A

=

A

⁢

A

∗

where

∗

denotes the

conjugate transpose

.

Similarly for a

real matrix

A

∈

ℝ

n

×

n

is said to be

normal

if

A

T

⁢

A

=

A

⁢

A

T

where

T

denotes the

transpose

.

properties:

•

Equivalently a complex matrix

A

∈

ℂ

n

×

n

is said to be

normal

if it satisfies

[

A

,

A

∗

]

=

0

where

[

,

]

is the

commutator bracket

.

•

Equivalently a real matrix

A

∈

ℝ

n

×

n

is said to be

normal

if it satisfies

[

A

,

A

T

]

=

0

where

[

,

]

is the commutator bracket.

•

Let

A

be a square complex matrix of order

n

. It follows from Schur’s inequality that if

A

is a

normal matrix

then

∑

i

=

1

n

|

λ

i

|

2

=

trace

⁡

A

∗

⁢

A

where

∗

is the conjugate transpose and

λ

i

are the

eigenvalues

of

A

.

•

A complex

square matrix

is

diagonal

if and only if it is normal, triangular.(see

theorem for normal triangular matrices

).

examples:

•

(

a

b

-

b

a

)

where

a

,

b

∈

ℝ

•

(

1

i

-

i

1

)

see also:

•

Wikipedia,

http://www.wikipedia.org/wiki/Normal_matrix

normal matrix

Title

normal matrix

Canonical name

NormalMatrix

Date of creation

2013-03-22 13:41:10

Last modified on

2013-03-22 13:41:10

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

12

Author

Daume (40)

Entry type

Definition

Classification

msc 15A21

Synonym

normal

Related topic

TheoremForNormalTriangularMatrices

Generated on Fri Feb 9 18:26:40 2018 by

LaTeXML
projection

projection

A

linear transformation

P

:

V

→

V

of a

vector space

V

is called a

projection

if it acts like the

identity

on its image. This
condition can be more succinctly expressed by the

equation

P

2

=

P

.

(1)

Proposition 1

If

P

:

V

→

V

is a projection, then
its image and the kernel are

complementary subspaces

, namely

V

=

ker

⁡

P

⊕

img

P

.

(2)

Proof.

Suppose that

P

is a projection. Let

v

∈

V

be given, and set

u

=

v

-

P

⁢

v

.

The projection condition (

1

) then implies
that

u

∈

ker

⁡

P

, and we can write

v

as the sum of an image and
kernel vectors:

v

=

u

+

P

⁢

v

.

This decomposition is unique, because the

intersection

of the image and the kernel is the

trivial

subspace

.
Indeed, suppose that

v

∈

V

is in both the image and the kernel of

P

.
Then,

P

⁢

v

=

v

and

P

⁢

v

=

0

, and hence

v

=

0

. QED

Conversely

, every

direct sum

decomposition

V

=

V

1

⊕

V

2

corresponds to a projection

P

:

V

→

V

defined by

P

⁢

v

=

{

v

v

∈

V

1

0

v

∈

V

2

Specializing somewhat, suppose that the ground field is

ℝ

or

ℂ

and that

V

is equipped with a positive-definite

inner
product

. In this setting we call an

endomorphism

P

:

V

→

V

an

orthogonal projection

if it is

self-dual

P

⋆

=

P

,

in

addition

to satisfying the projection condition (

1

).

Proposition 2

The kernel and image of an orthogonal projection are orthogonal subspaces.

Proof.

Let

u

∈

ker

⁡

P

and

v

∈

img

P

be given. Since

P

is self-dual we have

0

=

⟨

P

⁢

u

,

v

⟩

=

⟨

u

,

P

⁢

v

⟩

=

⟨

u

,

v

⟩

.

QED

Thus we see that a orthogonal projection

P

projects a

v

∈

V

onto

P

⁢

v

in an

orthogonal

fashion, i.e.

⟨

v

-

P

⁢

v

,

u

⟩

=

0

for all

u

∈

img

P

.

Title

projection

Canonical name

Projection

Date of creation

2013-03-22 12:52:13

Last modified on

2013-03-22 12:52:13

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

8

Author

rmilson (146)

Entry type

Definition

Classification

msc 15A21

Classification

msc 15A57

Defines

orthogonal projection

Generated on Fri Feb 9 18:26:42 2018 by

LaTeXML
theorem for normal matrices

theorem for normal matrices

Theorem

Let

A

be a

square matrix

. Then

A

is

normal matrix

if and only if there is a polynomial

g

such that

A

H

=

g

⁢

(

A

)

.

Title

theorem for normal matrices

Canonical name

TheoremForNormalMatrices

Date of creation

2013-03-22 15:29:24

Last modified on

2013-03-22 15:29:24

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

7

Author

georgiosl (7242)

Entry type

Theorem

Classification

msc 15A21

Generated on Fri Feb 9 18:26:58 2018 by

LaTeXML
commuting matrices are simultaneously triangularizable

commuting matrices are simultaneously triangularizable

Theorem 1

.

All matrices in the below are complex

n

×

n

matrices.

Let

A

,

B

be matrices and

A

⁢

B

=

B

⁢

A

. Then there exists a

unitary matrix

Q

such that

Q

H

⁢

A

⁢

Q

=

T

1

,

Q

H

⁢

B

⁢

Q

=

T

2

where

H

is the

conjugate transpose

and

T

1

,

T

2

,

are

upper triangular matrices

.

Title

commuting matrices are simultaneously triangularizable

Canonical name

CommutingMatricesAreSimultaneouslyTriangularizable

Date of creation

2013-03-22 15:26:48

Last modified on

2013-03-22 15:26:48

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

12

Author

georgiosl (7242)

Entry type

Theorem

Classification

msc 15A23

Related topic

SimultaneousUpperTriangularBlockDiagonalizationOfCommutingMatrices

Generated on Fri Feb 9 18:27:00 2018 by

LaTeXML
commuting normal matrices are simultaneously diagonalizable

commuting normal matrices are simultaneously diagonalizable

All matrices in the below are complex

n

×

n

matrices.

Let

A

,

B

be

normal matrices

,

A

⁢

B

=

B

⁢

A

. Then there exists a

unitary matrix

Q

such that

Q

H

⁢

A

⁢

Q

=

D

1

,

Q

H

⁢

B

⁢

Q

=

D

2

where

H

is the

conjugate transpose

and

D

1

,

D

2

are

diagonal matrices

.

Title

commuting normal matrices are simultaneously diagonalizable

Canonical name

CommutingNormalMatricesAreSimultaneouslyDiagonalizable

Date of creation

2013-03-22 15:26:51

Last modified on

2013-03-22 15:26:51

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

7

Author

georgiosl (7242)

Entry type

Corollary

Classification

msc 15A23

Generated on Fri Feb 9 18:27:02 2018 by

LaTeXML
matrix factorization

matrix factorization

Matrix Factorization

A

matrix factorization

(or

matrix decomposition

) is the right-hand-side product in

A

=

F

1

⁢

F

2

⁢

…

⁢

F

k

for “input” matrix

A

. The number of factor matrices

k

depends on the situation. Most often,

k

=

2

or

k

=

3

.

Note that the process of

producing

a factorization/

decomposition

is also called “factorization” or “decomposition”.

Examples

Some common factorizations and related devices are:

•

LU-decomposition

:

A

=

L

⁢

U

, where

L

is

lower triangular

, and

U

is upper triangular

•

QR-decomposition

:

A

=

Q

⁢

R

, where

Q

is

orthogonal

, and

R

is right triangular.

•

Singular value decomposition

(SVD):

A

=

U

⁢

S

⁢

V

T

, where

U

and

V

are orthogonal, and

S

is a partially

diagonal matrix

.

•

The

Cholesky Decomposition

.

•

For a

positive definite matrix

, we can decompose it into its

square root

(

http://planetmath.org/SquareRootOfPositiveDefiniteMatrix

)

squared

.

•

Polar decomposition

•

Jordan canonical form

•

Iwasawa decomposition

See the entries for these and other matrix factorizations for details on the contents of the factor matrices, where to apply them, and how to best calculate them.

Simultaneous matrix factorization

A related problem is to

diagonalize

or tridiagonalize many matrices using
the same matrix. Some results in this

direction

are listed below:

•

commuting matrices

are simultanenously triangularizable

•

commuting

normal matrices

are simultanenously

diagonalizable

Title

matrix factorization

Canonical name

MatrixFactorization

Date of creation

2013-03-22 14:15:07

Last modified on

2013-03-22 14:15:07

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

10

Author

mathcam (2727)

Entry type

Definition

Classification

msc 15A23

Synonym

matrix decomposition

Related topic

IsawasaDecomposition

Defines

factor matrix

Generated on Fri Feb 9 18:27:03 2018 by

LaTeXML
proof that commuting matrices are simultaneously triangularizable

proof that commuting matrices are simultaneously triangularizable

Proof by induction on

n

, order of matrix.

For

n

=

1

we can simply take

Q

=

1

.
We assume that there exists a common

unitary matrix

S

that triangularizes simultaneously

commuting matrices

,

(

n

-

1

)

×

(

n

-

1

)

.

So we have to show that the statement is valid for commuting matrices,

n

×

n

.
From

hypothesis

A

and

B

are commuting matrices

n

×

n

so these matrices have a common

eigenvector

.

Let

A

⁢

x

=

λ

⁢

x

,

B

⁢

x

=

μ

⁢

x

where

x

be the common eigenvector of unit length and

λ

,

μ

are the

eigenvalues

of

A

and

B

respectively. Consider the matrix,

R

=

(

x

X

)

where

X

be

orthogonal complement

of

x

and

R

H

⁢

R

=

I

, then we have that

R

H

⁢

A

⁢

R

=

(

λ

x

H

⁢

A

⁢

X

0

X

H

⁢

A

⁢

X

)

R

H

⁢

B

⁢

R

=

(

μ

x

H

⁢

B

⁢

X

0

X

H

⁢

B

⁢

X

)

It is

obvious

that the above matrices and also

X

H

⁢

B

⁢

X

,

X

H

⁢

A

⁢

X

,

(

n

-

1

)

×

(

n

-

1

)

matrices are commuting matrices. Let

B

1

=

X

H

⁢

B

⁢

X

and

A

1

=

X

H

⁢

A

⁢

X

then
there exists unitary matrix

S

such that

S

H

⁢

B

1

⁢

S

=

T

¯

2

,

S

H

⁢

A

1

⁢

S

=

T

¯

1

.

Now

Q

=

R

⁢

(

1

0

0

S

)

is a unitary matrix,

Q

H

⁢

Q

=

I

and we have

Q

H

⁢

A

⁢

Q

=

(

1

0

0

S

H

)

⁢

R

H

⁢

A

⁢

R

⁢

(

1

0

0

S

)

=

(

λ

x

H

⁢

A

⁢

X

⁢

S

0

T

¯

1

)

=

T

1

.

Analogously we have that

Q

H

⁢

B

⁢

Q

=

T

2

.

Title

proof that commuting matrices are simultaneously triangularizable

Canonical

name

ProofThatCommutingMatricesAreSimultaneouslyTriangularizable

Date of creation

2013-03-22 15:27:08

Last modified on

2013-03-22 15:27:08

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

10

Author

georgiosl (7242)

Entry type

Proof

Classification

msc 15A23

Generated on Fri Feb 9 18:27:05 2018 by

LaTeXML
spectral theorem

spectral theorem

Let

U

be a

finite-dimensional

,

unitary space

and let

M

:

U

→

U

be an

endomorphism

. We say that

M

is normal if it commutes with
its

Hermitian adjoint

, i.e.

M

⁢

M

⋆

=

M

⋆

⁢

M

.

Spectral Theorem

Let

M

:

U

→

U

be a

linear transformation

of a unitary space.
TFAE

1.

The

transformation

M

is normal.

2.

Letting

Λ

=

{

λ

∈

ℂ

∣

M

-

λ

⁢

E

⁢

is singular

}

,

where

E

is the

identity operator

,
denote the

spectrum

(set of

eigenvalues

) of

M

, the corresponding

eigenspaces

E

λ

=

ker

⁡

(

M

-

λ

⁢

E

)

,

λ

∈

Λ

give an

orthogonal

,

direct sum

decomposition of

U

, i.e.

U

=

⊕

λ

∈

Λ

E

λ

,

and

E

λ

1

⟂

E

λ

2

for distinct eigenvalues

λ

1

≠

λ

2

.

3.

We can decompose

M

as the sum

M

=

∑

λ

∈

Λ

λ

⁢

P

λ

,

where

Λ

∈

ℂ

is a finite subset of

complex numbers

indexing a family of commuting

orthogonal projections

P

λ

:

U

→

U

, i.e.

P

λ

=

⋆

P

λ

P

λ

P

μ

=

{

P

λ

λ

=

μ

0

λ

≠

μ

,

and where WLOG

∑

λ

∈

Λ

P

λ

=

1

U

.

4.

There exists an

orthonormal basis

of

U

that

diagonalizes

M

.

Remarks.

1.

Here are some important classes of

normal operators

,
distinguished by the nature of their eigenvalues.

–

Hermitian operators

. Eigenvalues are real.

–

Unitary transformations

. Eigenvalues lie on the

unit circle

,
i.e. the set of complex numbers of modulus 1.

–

Orthogonal projections. Eigenvalues are either 0 or 1.

2.

There is a well-known version of the

spectral theorem

for

ℝ

, namely that a

self-adjoint

(

symmetric

) transformation of a
real

inner product spaces

can diagonalized and that

eigenvectors

corresponding to different eigenvalues are orthogonal. An even more
down-to-earth version of this

theorem

says that a symmetric,

real
matrix

can always be diagonalized by an orthonormal basis of
eigenvectors.

3.

There are several versions of increasing sophistication of the
spectral theorem that hold in infinite-dimensional,

Hilbert space

setting. In such a

context

one must distinguish between the
so-called

discrete

and

continuous

(no corresponding eigenspace)
spectrums, and replace the representing sum for the

operator

with
some kind of an integral. The definition of self-adjointness is also
quite tricky for

unbounded operators

. Finally, there are versions
of the spectral theorem, of importance in theoretical quantum
mechanics, that can be applied to continuous 1-parameter groups of
commuting, self-adjoint operators.

Title

spectral theorem

Canonical name

SpectralTheorem

Date of creation

2013-03-22 12:45:49

Last modified on

2013-03-22 12:45:49

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

9

Author

rmilson (146)

Entry type

Theorem

Classification

msc 15A23

Classification

msc 15A63

Classification

msc 15A18

Related topic

DiagonalizableOperator

Defines

normal operator

Generated on Fri Feb 9 18:27:07 2018 by

LaTeXML
centralizer of matrix units

centralizer of matrix units

Theorem -

Let

R

be a

ring with identity

1

and

M

n

⁢

(

R

)

the ring of

n

×

n

matrices with entries in

R

. The

centralizer

of the

matrix units

is the set

R

⋅

I

⁢

d

, consisting of all

multiples

of the

identity matrix

.

:

It is clear that the multiples of the identity matrix commute with all matrix units, and therefore belong to their centralizer. We will now prove the converse.

We will regard the elements of

M

n

⁢

(

R

)

as

endomorphisms

of the module

⊕

i

=

1

n

R

. We will denote by

{

e

i

}

the canonical basis of

⊕

i

=

1

n

R

and by

E

i

⁢

j

the matrix unit whose entry

(

i

,

j

)

is

1

.

Let

S

=

[

s

i

⁢

j

]

∈

M

n

⁢

(

R

)

be an element of the centralizer of the matrix units. For all

i

,

j

,

k

we must have

S

⁢

E

i

⁢

j

⁢

e

k

=

E

i

⁢

j

⁢

S

⁢

e

k

(1)

But a straightforward computation shows that

S

⁢

E

i

⁢

j

⁢

e

j

=

S

⁢

e

i

and

E

i

⁢

j

⁢

S

⁢

e

j

=

s

j

⁢

j

⁢

e

i

. Since

j

is arbitrary we see, by equality (1), that all

s

j

⁢

j

are equal, say

s

j

⁢

j

=

s

∈

R

.

Hence,

S

⁢

e

i

=

s

⁢

e

i

, wich means that

S

=

s

⁢

I

⁢

d

. We conclude that

S

must be a multiple of the identity matrix.

□

Title

centralizer of matrix units

Canonical name

CentralizerOfMatrixUnits

Date of creation

2013-03-22 18:39:58

Last modified on

2013-03-22 18:39:58

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

8

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 15A30

Classification

msc 16S50

Generated on Fri Feb 9 18:27:09 2018 by

LaTeXML
Gerstenhaber - Serezhkin theorem

Gerstenhaber - Serezhkin theorem

Let

𝔽

be an arbitrary field. Consider

ℳ

n

⁢

(

𝔽

)

,

the

vector space

of all

n

×

n

matrices over

𝔽

.

Define

•

𝒩

=

{

A

∈

ℳ

n

⁢

(

𝔽

)

:

A

⁢

is nilpotent

}

,

•

𝒢

⁢

ℒ

n

⁢

(

𝔽

)

=

{

A

∈

ℳ

n

⁢

(

𝔽

)

:

det

⁡

(

A

)

≠

0

}

,

•

𝒯

=

{

A

∈

ℳ

n

⁢

(

𝔽

)

:

A

⁢

is strictly upper triangular

}

.

Notice that

𝒯

is a

linear subspace

of

ℳ

n

⁢

(

𝔽

)

.

Moreover,

𝒯

⊆

𝒩

and

dim

⁡

𝒯

=

n

⁢

(

n

-

1

)

/

2

.

The Gerstenhaber – Serezhkin theorem on linear subspaces contained in the nilpotent cone

[

G

,

S

]

reads as follows.

Theorem 1

Let

L

be a linear subspace of

M

n

⁢

(

F

)

.

Assume that

L

⊆

N

.

Then

(i)

dim

⁡

L

≤

n

⁢

(

n

-

1

)

/

2

,

(ii)

dim

⁡

L

=

n

⁢

(

n

-

1

)

/

2

if and only if there exists

U

∈

G

⁢

L

n

⁢

(

F

)

such that

{

U

⁢

A

⁢

U

-

1

:

A

∈

L

}

=

T

.

An alternative simple proof of inequality

(i)

can be found in

[

M

]

.

References

G

M. Gerstenhaber, On nilalgebras and linear

varieties

of

nilpotent matrices

, I,

Amer. J. Math.

80

: 614–622 (1958).

M

B. Mathes, M. Omladič, H. Radjavi,

Linear Spaces

of Nilpotent Matrices,

Linear Algebra

Appl.

149

: 215–225 (1991).

S

V. N. Serezhkin, On

linear transformations

preserving nilpotency,

Vests

ι

¯

Akad. Navuk BSSR Ser. F

ι

¯

z.-Mat. Navuk

1985

, no. 6: 46–50 (Russian).

Title

Gerstenhaber - Serezhkin theorem

Canonical name

GerstenhaberSerezhkinTheorem

Date of creation

2013-03-22 19:20:05

Last modified on

2013-03-22 19:20:05

Owner

kammerer (26336)

Last modified by

kammerer (26336)

Numerical id

7

Author

kammerer (26336)

Entry type

Theorem

Classification

msc 15A30

Related topic

BottaPierceWatkinsTheorem

Generated on Fri Feb 9 18:27:14 2018 by

LaTeXML
ideals in matrix algebras

ideals in matrix algebras

Let

R

be a ring with 1. Consider the ring

M

n

×

n

⁢

(

R

)

of

n

×

n

-matrices with entries taken from

R

.

It will be shown that there exists a

one-to-one
correspondence

between the (two-sided) ideals of

R

and the
(two-sided) ideals of

M

n

×

n

⁢

(

R

)

.

For

1

≤

i

,

j

≤

n

, let

E

i

⁢

j

denote the

n

×

n

-matrix having entry 1 at position

(

i

,

j

)

and 0 in all other places. It can be
easily checked that

E

i

⁢

j

⋅

E

k

⁢

l

=

{

0

iff

k

≠

j

E

i

⁢

l

otherwise.

(1)

Let

𝔪

be an ideal in

M

n

×

n

⁢

(

R

)

.

Claim

.

The set

i

⊆

R

given by

𝔦

=

{

x

∈

R

∣

x

is an entry
of

⁢

A

∈

𝔪

}

is an ideal in

R

, and

m

=

M

n

×

n

⁢

(

i

)

.

Proof.

𝔦

≠

∅

since

0

∈

𝔦

. Now let

A

=

(

a

i

⁢

j

)

and

B

=

(

b

i

⁢

j

)

be matrices in

𝔪

, and

x

,

y

∈

R

be
entries of

A

and

B

respectively, say

x

=

a

i

⁢

j

and

y

=

b

k

⁢

l

. Then the matrix

A

⋅

E

j

⁢

l

+

E

i

⁢

k

⋅

B

∈

𝔪

has

x

+

y

at position

(

i

,

l

)

, and it follows: If

x

,

y

∈

𝔦

, then

x

+

y

∈

𝔦

. Since

𝔦

is an ideal in

M

n

×

n

⁢

(

R

)

it contains, in particular, the matrices

D

r

⋅

A

and

A

⋅

D

r

, where

D

r

:=

∑

i

=

1

n

r

⋅

E

i

⁢

i

,

r

∈

R

.

thus,

r

⁢

x

,

x

⁢

r

∈

𝔦

. This shows
that

𝔦

is an ideal in

R

.
Furthermore,

M

n

×

n

⁢

(

𝔦

)

⊆

𝔪

.

By construction, any matrix

A

∈

𝔪

has entries in

𝔦

, so we have

A

=

∑

1

≤

i

,

j

≤

n

a

i

⁢

j

⁢

E

i

⁢

j

,

a

i

⁢

j

∈

𝔦

so

A

∈

m

n

×

n

⁢

(

𝔦

)

. Therefore

𝔪

⊆

M

n

×

n

⁢

(

𝔦

)

.
∎

A

consequence

of this is: If

F

is a field, then

M

n

×

n

⁢

(

F

)

is simple.

Title

ideals in matrix algebras

Canonical name

IdealsInMatrixAlgebras

Date of creation

2013-03-22 13:59:28

Last modified on

2013-03-22 13:59:28

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

10

Author

mathcam (2727)

Entry type

Topic

Classification

msc 15A30

Generated on Fri Feb 9 18:27:16 2018 by

LaTeXML