aarjaneiro/planetmath huggingface dataset:
underlying graph of a quiver

underlying graph of a quiver

Let

Q

=

(

Q

0

,

Q

1

,

s

,

t

)

be a quiver, i.e.

Q

0

is a set of vertices,

Q

1

is a set of arrows and

s

,

t

:

Q

1

→

Q

0

are

functions

which take each arrow to its source and target respectively.

Definition.

An

underlying graph of

Q

or

graph associated with

Q

is a graph

G

=

(

V

,

E

,

τ

)

such that

V

=

Q

0

,

E

=

Q

1

and

τ

:

E

→

V

s

⁢

y

⁢

m

2

is given by

τ

⁢

(

α

)

=

[

s

⁢

(

α

)

,

t

⁢

(

α

)

]

∼

.

In other words

G

is a graph which is obtained from

Q

after forgeting the

orientation

of arrows. The definition of a graph used here is taken from this entry (

http://planetmath.org/AlternativeDefinitionOfAMultigraph

).

Note, that if we know the underlying graph

G

of a quiver

Q

, then the information we have is not enough to reconstruct

Q

(except for a

trivial

case with no edges). The orientation of arrows is lost forever. In some cases it is possible to reconstruct

Q

up to an

isomorphism

of quivers (

http://planetmath.org/MorphismsBetweenQuivers

), for example graph

\xymatrix

⁢

1

⁢

\ar

⁢

@

-

[

r

]

⁢

&

⁢

2

uniquely (up to isomorphism) determines its quiver, but

\xymatrix

⁢

G

:

&

⁢

1

⁢

\ar

⁢

@

-

[

r

]

⁢

&

⁢

2

⁢

\ar

⁢

@

-

[

r

]

⁢

&

⁢

3

does not uniquely determine its quiver. Indeed, there are exactly two nonisomorphic quivers with underlying graph

G

, namely:

\xymatrix

⁢

Q

:

&

⁢

1

⁢

\ar

⁢

[

r

]

⁢

&

⁢

2

⁢

\ar

⁢

[

r

]

⁢

&

⁢

3

⁢

Q

′

:

&

⁢

1

⁢

\ar

⁢

[

r

]

⁢

&

⁢

2

⁢

&

⁢

3

⁢

\ar

⁢

[

l

]

Title

underlying graph of a quiver

Canonical name

UnderlyingGraphOfAQuiver

Date of creation

2013-03-22 19:16:59

Last modified on

2013-03-22 19:16:59

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Definition

Classification

msc 14L24

Generated on Fri Feb 9 18:00:33 2018 by

LaTeXML
unital path algebras

unital path algebras

Let

Q

be a quiver and

k

an arbitrary field.

Proposition

.

The path

algebra

k

⁢

Q

is

unitary

if and only if

Q

has a finite number of vertices.

Proof.

,,

⇒

” Assume, that

Q

has an

infinite

number of vertices and let

1

∈

k

⁢

Q

be an

identity

. Then we can express

1

as

1

=

∑

i

=

1

n

λ

n

⋅

w

n

where

λ

n

∈

k

and

w

n

are paths (they form a basis of

k

⁢

Q

as a

vector space

). Since

Q

has an infinite number of vertices, then we can take a

stationary

path

e

x

for some vertex

x

such that there is no path among

w

1

,

…

,

w

n

ending in

x

. By definition of

k

⁢

Q

and by the fact that

1

is an identity we have:

e

x

=

1

⋅

e

x

=

(

∑

i

=

1

n

λ

n

⋅

w

n

)

⋅

e

x

=

∑

i

=

1

n

λ

n

⋅

(

w

n

⋅

e

x

)

=

∑

i

=

1

n

λ

n

⋅

0

=

0

.

Contradiction

.

□

,,

⇐

” If the set

Q

0

of vertices of

Q

is finite, then put

1

=

∑

q

∈

Q

0

e

q

where

e

q

denotes the stationary path (note that

1

is well-defined, since the sum is finite). If

w

is a path in

Q

from

x

to

y

, then

e

x

⋅

w

=

w

and

w

⋅

e

y

=

w

. All other

combinations

of

w

with

e

q

yield

0

and thus we obtain that

1

⋅

w

=

w

=

w

⋅

1

.

This

completes

the proof.

□

Title

unital path algebras

Canonical name

UnitalPathAlgebras

Date of creation

2013-03-22 19:16:23

Last modified on

2013-03-22 19:16:23

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Theorem

Classification

msc 14L24

Generated on Fri Feb 9 18:00:35 2018 by

LaTeXML
normal complex analytic variety

normal complex analytic variety

Let

V

be a local

complex analytic variety

(or a

complex analytic space

). A point

p

∈

V

is if and only if every

weakly holomorphic

function

through

V

extends to be

holomorphic

in

V

near

p

.

In particular, if

V

⊂

ℂ

n

is a complex analytic subvariety, it is normal at

p

if and only if every weakly holomorphic function through

V

extends to be holomorphic in a neighbourhood of

p

in

ℂ

n

.

To see that this definition is equivalent to the usual one, that is, that

V

is normal
at

p

if and only if

𝒪

p

(the ring of germs of

holomorphic functions

at

p

)
is

integrally closed

, we need the following theorem. Let

ℳ

p

be the total quotient ring of

𝒪

p

, that is, the ring of germs of

meromorphic functions

.

Theorem

.

Let

V

be a local complex analytic variety. Then

O

p

w

⁢

(

V

)

is the

integral
closure

of

O

p

⁢

(

V

)

in

M

p

.

References

1

Hassler Whitney.
.
Addison-Wesley, Philippines, 1972.

Title

normal complex analytic variety

Canonical name

NormalComplexAnalyticVariety

Date of creation

2013-03-22 17:41:48

Last modified on

2013-03-22 17:41:48

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Definition

Classification

msc 14M05

Classification

msc 32C20

Synonym

normal analytic variety

Related topic

WeaklyHolomorphic

Defines

normal complex analytic space

Defines

normal complex analytic subvariety

Defines

normal analytic space

Defines

normal analytic subvariety

Generated on Fri Feb 9 18:00:36 2018 by

LaTeXML
normal variety

normal variety

Let

X

be a

variety

.

X

is said to be

normal

at a point

p

∈

X

if the

local ring

𝒪

p

is

integrally closed

.

X

is said to be

normal

if it is normal at every point. If

X

is

non-singular

at

p

, it is normal at

p

, since

regular local rings

are integrally closed.

Title

normal variety

Canonical name

NormalVariety

Date of creation

2013-03-22 13:20:28

Last modified on

2013-03-22 13:20:28

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Definition

Classification

msc 14M05

Generated on Fri Feb 9 18:00:38 2018 by

LaTeXML
flag variety

flag variety

Let

k

be a field, and let

V

be a

vector space

over

k

of

dimension

n

and
choose an increasing sequence

𝐢

=

(

i

1

,

…

,

i

m

)

, with

1

≤

i

1

<

⋯

<

i

m

≤

n

. Then the

(partial)

flag variety

ℱ

⁢

ℓ

⁢

(

V

,

𝐢

)

associated to this data is the set of all
flags

{

0

}

≤

V

1

⊂

⋯

⊂

V

n

with

dim

⁡

V

j

=

i

j

. This has a

natural embedding

into the product of Grassmannians

G

⁢

(

V

,

i

1

)

×

⋯

⁢

G

⁢

(

V

,

i

m

)

, and its image here is closed, making

ℱ

⁢

ℓ

⁢

(

V

,

𝐢

)

into a

projective variety

over

k

. If

k

=

ℂ

these are often called

flag manifolds

.

The group

Sl

⁢

(

V

)

acts transtively on

ℱ

⁢

ℓ

⁢

(

V

,

𝐢

)

,
and the

stabilizer

of a point is a

parabolic subgroup

. Thus, as a

homogeneous
space

,

ℱ

⁢

ℓ

⁢

(

V

,

𝐢

)

≅

Sl

⁢

(

V

)

/

P

where

P

is a parabolic
subgroup of

Sl

⁢

(

V

)

. In particular, the complete flag variety is

isomorphic

to

Sl

⁢

(

V

)

/

B

, where

B

is the

Borel subgroup

.

Title

flag variety

Canonical name

FlagVariety

Date of creation

2013-03-22 13:27:56

Last modified on

2013-03-22 13:27:56

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

6

Author

bwebste (988)

Entry type

Definition

Classification

msc 14M15

Synonym

flag manifold

Defines

complete flag variety

Defines

partial flag variety

Generated on Fri Feb 9 18:00:43 2018 by

LaTeXML
Cartan’s umbrella

Cartan’s umbrella

The

Cartan’s umbrella

refers to a certain class of examples of

real analytic sets

(in fact real

algebraic

usually) in

ℝ

3

,
which are

irreducible

(not written as a union of

proper subsets

that are also

subvarieties

), and where
there are regular points both of

dimension

1 and of dimension 2. Sometimes higher dimensional examples with

similar

behavior are also called the same. A fairly common equation for a

Cartan umbrella

is

z

⁢

(

x

2

+

y

2

)

-

y

3

=

0

.

Solving for

z

we get

z

=

y

3

x

2

+

y

2

. The graph of this

function

is shown in the following figure.

Figure 1: Graph of

z

=

y

3

x

2

+

y

2

The umbrella itself also includes the

z

axis, since all points of the form

(

0

,

0

,

z

)

satisfy

the equation. . It is impossible to write an equation (real analytic or real algebraic) whose solution set contains the graph in Figure 1, and such that the

z

axis is not included.

This

pathological

behavior does not happen for complex analytic subvarieties.

References

1

Jacek Bochnak, Michel Coste, Marie-Francoise Roy.
.
Springer, 1998.

Title

Cartan’s umbrella

Canonical

name

CartansUmbrella

Date of creation

2013-03-22 17:41:11

Last modified on

2013-03-22 17:41:11

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

4

Author

jirka (4157)

Entry type

Example

Classification

msc 14P05

Classification

msc 14P15

Synonym

Cartan umbrella

Generated on Fri Feb 9 18:01:06 2018 by

LaTeXML
real analytic subvariety

real analytic subvariety

Let

U

⊂

ℝ

N

be an open set.

Definition

.

A closed set

X

⊂

U

is called a

real analytic subvariety

of

U

such that for each point

p

∈

X

, there exists a neigbourhood

V

and a set

ℱ

of

real analytic functions

defined in

V

, such that

X

∩

V

=

{

p

∈

V

∣

f

⁢

(

p

)

=

0

⁢

for all

⁢

f

∈

ℱ

}

.

If

U

=

ℝ

N

and all the

f

∈

ℱ

are real polynomials, then

X

is said to be a

real algebraic subvariety

.

If

X

is not required to be closed, then it is said to be a

local real analytic subvariety

.
Sometimes

X

is called a real analytic set or real analytic variety. Similarly as for

complex
analytic

sets we can also define the regular and

singular points

.

Definition

.

A point

p

∈

X

is called a

regular point

if there is a neighbourhood

V

of

p

such that

X

∩

V

is a

submanifold

. Any other
point is called a

singular point

.

The set of regular points of

X

is denoted by

X

-

or sometimes

X

*

.

The set of singular points
is no longer a

subvariety

as in the

complex

case, though it can be sown to be

semianalytic

. In general, real subvarieties is far worse behaved than their complex counterparts.

References

1

Jacek Bochnak, Michel Coste, Marie-Francoise Roy.
.
Springer, 1998.

Title

real analytic subvariety

Canonical

name

RealAnalyticSubvariety

Date of creation

2013-03-22 17:41:07

Last modified on

2013-03-22 17:41:07

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

4

Author

jirka (4157)

Entry type

Definition

Classification

msc 14P05

Classification

msc 14P15

Synonym

real analytic variety

Synonym

real analytic set

Related topic

SmoothSubmanifoldContainedInASubvarietyOfSameDimensionIsRealAnalytic

Defines

real algebraic variety

Defines

real algebraic subvariety

Defines

local real analytic subvariety

Defines

regular point

Defines

singular point

Generated on Fri Feb 9 18:01:08 2018 by

LaTeXML
semialgebraic set

semialgebraic set

Definition

.

Consider the

A

⊂

ℝ

n

,
defined by
real polynomials

p

j

⁢

ℓ

,

j

=

1

,

…

,

k

,

ℓ

=

1

,

…

,

m

,
and the

relations

ϵ

j

⁢

ℓ

where

ϵ

j

⁢

ℓ

is

>

,

=

, or

<

.

A

=

⋃

ℓ

=

1

m

{

x

∈

ℝ

n

∣

p

j

⁢

ℓ

⁢

(

x

)

⁢

ϵ

j

⁢

ℓ

⁢

0

,

j

=

1

,

…

,

k

}

.

(1)

Sets of this form are said to be

semialgebraic

.

Similarly as

algebraic

subvarieties

, finite union and

intersection

of

semialgebraic sets

is still a semialgebraic set. Furthermore, unlike subvarieties, the

complement

of a semialgebraic set is again semialgebraic. Finally, and most importantly, the

Tarski-Seidenberg theorem

says that they are also

closed under

projection

.

On a dense open subset of

A

,

A

is (locally) a

submanifold

, and hence we can easily
define the

dimension

of

A

to be the largest dimension at points at which

A

is a submanifold. It is not hard to see that a semialgebraic set
lies inside an algebraic subvariety of the same dimension.

References

1

Edward Bierstone and Pierre D. Milman,

Semianalytic

and subanalytic
sets

, Inst. Hautes Études Sci. Publ. Math. (1988), no. 67, 5–42.

http://www.ams.org/mathscinet-getitem?mr=89k:32011

MR 89k:32011

Title

semialgebraic set

Canonical name

SemialgebraicSet

Date of creation

2013-03-22 16:46:10

Last modified on

2013-03-22 16:46:10

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Definition

Classification

msc 14P10

Related topic

TarskiSeidenbergTheorem

Related topic

SubanalyticSet

Defines

semialgebraic

Defines

dimension of a semialgebraic set

Generated on Fri Feb 9 18:01:09 2018 by

LaTeXML
Tarski-Seidenberg theorem

Tarski-Seidenberg theorem

Theorem

(Tarski-Seidenberg).

The set of

semialgebraic sets

is

closed under

projection

.

That is, if

A

⊂

ℝ

n

×

ℝ

m

is a semialgebraic set, and if

π

is the projection onto the first

n

coordinates

, then

π

⁢

(

A

)

is also semialgebraic.

Łojasiewicz generalized this

theorem

further. For this we need a bit of notation.

Let

U

⊂

ℝ

n

.
Suppose

𝒜

⁢

(

U

)

is any ring of real valued

functions

on

U

.
Define

𝒮

⁢

(

𝒜

⁢

(

U

)

)

to be the smallest
set of subsets of

U

, which contain the sets

{

x

∈

U

⁢

∣

f

⁢

(

x

)

>

⁢

0

}

for all

f

∈

𝒜

⁢

(

U

)

,
and is closed under finite union, finite

intersection

and

complement

.
Let

𝒜

⁢

(

U

)

⁢

[

t

]

denote the ring of polynomials in

t

∈

ℝ

m

with coefficients in

𝒜

⁢

(

U

)

.

Theorem

(Tarski-Seidenberg-Łojasiewicz).

Suppose that

V

⊂

U

×

R

m

⊂

R

n

+

m

,
is such that

V

∈

S

⁢

(

A

⁢

(

U

)

⁢

[

t

]

)

.
Then the projection of

V

onto the first

n

variables
is in

S

⁢

(

A

⁢

(

U

)

)

.

References

1

Edward Bierstone and Pierre D. Milman,

Semianalytic

and subanalytic
sets

, Inst. Hautes Études Sci. Publ. Math. (1988), no. 67, 5–42.

http://www.ams.org/mathscinet-getitem?mr=89k:32011

MR 89k:32011

Title

Tarski-Seidenberg theorem

Canonical name

TarskiSeidenbergTheorem

Date of creation

2013-03-22 16:46:13

Last modified on

2013-03-22 16:46:13

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Theorem

Classification

msc 14P15

Classification

msc 14P10

Related topic

SemialgebraicSet

Related topic

SubanalyticSet

Defines

Tarski-Seidenberg-Łojasiewicz theorem

Generated on Fri Feb 9 18:01:11 2018 by

LaTeXML
algebraic manifold

algebraic manifold

Definition

.

Let

k

be a field and let

M

⊂

k

n

be a

submanifold

.

M

is said to
be an

algebraic manifold

(or

k

-

algebraic

) if there exists an

irreducible

algebraic
variety

V

⊂

k

n

such that

dim

⁡

V

=

dim

⁡

M

and

M

⊂

V

. If

k

=

ℝ

,
then

M

is called a

Nash manifold

.

It can be proved that such a

manifold

is defined as the

zero set

of a finite collection of

analytic algebraic functions

.

References

1

M. Salah Baouendi,
Peter Ebenfelt,
Linda Preiss Rothschild.
,
Princeton University Press,
Princeton, New Jersey, 1999.

Title

algebraic manifold

Canonical name

AlgebraicManifold

Date of creation

2013-03-22 15:36:08

Last modified on

2013-03-22 15:36:08

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

6

Author

jirka (4157)

Entry type

Definition

Classification

msc 14P20

Classification

msc 14-00

Classification

msc 58A07

Synonym

algebraic submanifold

Synonym

k

-algebraic manifold

Synonym

k

-algebraic submanifold

Defines

Nash manifold

Defines

Nash submanifold

Generated on Fri Feb 9 18:01:13 2018 by

LaTeXML
smooth submanifold contained in a subvariety of same dimension is real analytic

smooth submanifold contained in a subvariety of same dimension is real analytic

This theorem seems to usually be attributed to Malgrange in literature as it appeared in his book

[

1

]

.

Theorem

(Malgrange).

Suppose

M

⊂

R

N

is a

connected

smooth (

C

∞

)

submanifold

and

V

⊂

R

N

is a

real analytic
subvariety

of the same dimension as

M

, such that

M

⊂

V

. Then

M

is a real analytic submanifold.

The condition that

M

is smooth cannot be relaxed to

C

k

for

k

<

∞

. For example, note
that in

ℝ

2

, the

subvariety

y

3

-

x

8

=

0

, which is the graph of the

C

1

function

y

=

|

x

|

8

3

, is not a real analytic submanifold.

References

1

Bernard Malgrange.
.
Oxford University Press, 1966.

Title

smooth submanifold contained in a subvariety of same dimension is real analytic

Canonical name

SmoothSubmanifoldContainedInASubvarietyOfSameDimensionIsRealAnalytic

Date of creation

2013-03-22 17:41:16

Last modified on

2013-03-22 17:41:16

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Theorem

Classification

msc 14P99

Related topic

RealAnalyticSubvariety

Generated on Fri Feb 9 18:01:15 2018 by

LaTeXML
affine space

affine space

Definition

.

Let

K

be a field and let

n

be a

positive integer

. In

algebraic geometry

we define

affine space

(or affine

n

-space) to be the set

{

(

k

1

,

…

,

k

n

)

:

k

i

∈

K

}

.

Affine space is usually denoted by

K

n

or

A

n

(or

A

n

⁢

(

K

)

if we want to emphasize the field of definition).

In Algebraic Geometry, we consider affine space as a topological space, with the usual

Zariski topology

(see also

algebraic set

,

affine variety

). The

polynomials

in the ring

K

⁢

[

x

1

,

…

,

x

n

]

are regarded as functions (

algebraic functions

) on

𝔸

n

⁢

(

K

)

. “Gluing” several copies of affine space one obtains a

projective space

.

Lemma

.

If

K

is

algebraically closed

, affine space

A

n

⁢

(

K

)

is an

irreducible

algebraic variety

.

References

1

R. Hartshorne,

Algebraic Geometry

,
Springer-Verlag, New York.

Title

affine space

Canonical name

AffineSpace

Date of creation

2013-03-22 15:14:21

Last modified on

2013-03-22 15:14:21

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

8

Author

alozano (2414)

Entry type

Definition

Classification

msc 14R10

Classification

msc 14-00

Related topic

ProjectiveSpace

Related topic

AffineVariety

Generated on Fri Feb 9 18:01:17 2018 by

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Jacobian conjecture

Jacobian conjecture

Let

F

:

ℂ

n

→

ℂ

n

be a polynomial map, i.e.,

F

⁢

(

x

1

,

…

,

x

n

)

=

(

f

1

⁢

(

x

1

,

…

,

x

n

)

,

…

,

f

n

⁢

(

x

1

,

…

,

x

n

)

)

for certain

polynomials

f

i

∈

ℂ

⁢

[

X

1

,

…

,

X

n

]

.

If

F

is

invertible

, then its

Jacobi determinant

det

⁡

(

∂

⁡

f

i

/

∂

⁡

x

j

)

, which is a polynomial over

ℂ

,
vanishes nowhere and hence must be a non-zero

constant

.

The

Jacobian conjecture

asserts the

converse

: every polynomial map

ℂ

n

→

ℂ

n

whose Jacobi determinant is a non-zero constant
is invertible.

Title

Jacobian conjecture

Canonical name

JacobianConjecture

Date of creation

2013-03-22 13:23:46

Last modified on

2013-03-22 13:23:46

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Conjecture

Classification

msc 14R15

Synonym

Keller’s problem

Generated on Fri Feb 9 18:01:19 2018 by

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an example for Schur decomposition

an example for Schur decomposition

Let

A

=

(

5

7

-

2

-

4

)

.

We will find an

orthogonal matrix

P

and an

upper triangular matrix

T

such that

P

t

⁢

A

⁢

P

=

T

applying the proof of Schur’s

decomposition

.
We ’re following the steps below

•

We find the

eigenvalues

of

A

The eigenvalues of a matrix are precisely the solutions to the equation

det

⁡

(

λ

⁢

I

-

A

)

=

0

↔

λ

2

-

λ

-

6

=

0

Hence the roots of the

quadratic equation

(

http://planetmath.org/QuadraticFormula

) are the eigenvalues

λ

1

=

-

2

,

λ

2

=

3

•

We find the

eigenvectors

For each eigenvalue

λ

i

, solving the system

(

A

-

λ

i

⁢

I

)

⁢

X

i

=

0

So we have that
for

λ

1

=

-

2

(

A

+

2

⁢

I

)

=

0

↔

(

7

7

-

2

-

2

)

⁢

(

x

1

x

2

)

=

(

0

0

)

→

X

1

=

(

1

,

-

1

)

Analogously for

λ

2

=

3

the eigenvector

X

2

=

(

7

,

-

2

)

•

We get an

orthonormal set

of eigenvectors using

Gram-Schmidt orthogonalization

Consider the above two eigenvectors which are

linearly independent

but are not

orthogonal

X

1

=

(

1

,

-

1

)

X

2

=

(

7

,

-

2

)

First we take

w

1

=

X

1

=

(

1

,

-

1

)

. Therefore

w

2

=

X

2

-

w

1

⋅

X

2

∥

w

1

∥

2

⁢

w

1

that is,

w

2

=

(

5

2

,

5

2

)

and finally the orthonormal set is

{

w

1

/

∥

w

1

∥

,

w

2

/

∥

w

2

∥

}

=

{

(

1

2

,

-

1

2

)

,

(

1

2

,

1

2

)

}

So

P

=

1

2

⁢

(

1

1

-

1

1

)

.

Then

T

=

P

t

⁢

A

⁢

P

=

(

-

2

9

0

3

)

.

Title

an example for

Schur decomposition

Canonical name

AnExampleForSchurDecomposition

Date of creation

2013-03-22 15:27:02

Last modified on

2013-03-22 15:27:02

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

8

Author

georgiosl (7242)

Entry type

Application

Classification

msc 15-00

Related topic

SchurDecomposition

Related topic

GramSchmidtOrthogonalization

Generated on Fri Feb 9 18:13:36 2018 by

LaTeXML
anti-diagonal matrix

anti-diagonal matrix

Let

A

be a

square matrix

(over any field

𝔽

). An entry in

A

is an

anti-diagonal entry

if it is on the line going from the
lower left corner of

A

to the upper right corner. If all
entries in

A

are zero except on the anti-diagonal, then

A

is an

anti-diagonal matrix

.

If

a

1

,

…

,

a

n

∈

𝔽

, let

adiag

⁡

(

a

1

,

…

,

a

n

)

=

(

0

0

0

0

a

1

0

0

0

a

2

0

0

0

a

3

0

0

⋅

0

0

a

n

0

0

0

)

.

Properties of anti-diagonal matrices

1.

If

A

and

D

are

n

×

n

anti-diagonal and

diagonal matrices

, respectively, then

A

⁢

D

,

D

⁢

A

are anti-diagonal.

2.

The product of two anti-diagonal matrices is an diagonal matrix.

Title

anti-diagonal matrix

Canonical name

AntidiagonalMatrix

Date of creation

2013-03-22 15:12:20

Last modified on

2013-03-22 15:12:20

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

8

Author

matte (1858)

Entry type

Definition

Classification

msc 15-00

Generated on Fri Feb 9 18:13:38 2018 by

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characteristic polynomial of a orthogonal matrix is a reciprocal polynomial

characteristic polynomial of a orthogonal matrix is a reciprocal polynomial

Theorem 1

.

The

characteristic polynomial

of a

orthogonal matrix

is a

reciprocal polynomial

Proof.

Let

A

be the orthogonal matrix, and let

p

⁢

(

λ

)

=

det

⁡

(

A

-

λ

⁢

I

)

be
its characteristic polynomial. We wish to prove that

p

⁢

(

λ

)

=

±

λ

n

⁢

p

⁢

(

1

/

λ

)

.

Since

A

-

1

=

A

T

, we have

A

-

λ

⁢

I

=

-

λ

⁢

A

⁢

(

A

T

-

I

/

λ

)

.

Taking the

determinant

of both sides, and using

det

⁡

A

=

det

⁡

A

T

and

det

⁡

c

⁢

A

=

c

n

⁢

det

⁡

A

(

c

∈

ℂ

),
yields

det

⁡

(

A

-

λ

⁢

I

)

=

±

λ

n

⁢

det

⁡

(

A

-

1

λ

⁢

I

)

.

∎

References

1

H. Eves,

Elementary Matrix

Theory

, Dover publications, 1980.

Title

characteristic polynomial of a orthogonal matrix is a reciprocal polynomial

Canonical name

CharacteristicPolynomialOfAOrthogonalMatrixIsAReciprocalPolynomial

Date of creation

2013-03-22 15:33:13

Last modified on

2013-03-22 15:33:13

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

5

Author

matte (1858)

Entry type

Theorem

Classification

msc 15-00

Related topic

CharacteristicPolynomialOfASymplecticMatrixIsAReciprocalPolynomial

Generated on Fri Feb 9 18:13:40 2018 by

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