aarjaneiro/planetmath huggingface dataset:
proof of Cauchy-Schwarz inequality

proof of Cauchy-Schwarz inequality

If

a

and

b

are

linearly dependent

, we write

𝒃

=

λ

⁢

𝒂

. So we get:

⟨

𝒂

,

λ

⁢

𝒂

⟩

2

=

λ

2

⁢

⟨

𝒂

,

𝒂

⟩

2

=

λ

2

⁢

||

𝒂

||

4

=

||

𝒂

||

2

⁢

||

𝒃

||

2

.

So we have equality if

𝒂

and

𝒃

are linearly dependent. In the other case we look at the quadratic function

||

x

⋅

𝒂

+

𝒃

||

2

=

x

2

⁢

||

𝒂

||

2

+

2

⁢

x

⁢

⟨

𝒂

,

𝒃

⟩

+

||

𝒃

||

2

.

This function is positive for every real

x

, if

𝒂

and

𝒃

are

linearly independent

. Thus it has no real zeroes, which means that

⟨

𝒂

,

𝒃

⟩

2

-

||

𝒂

||

2

⁢

||

𝒃

||

2

is always negative. So we have:

⟨

𝒂

,

𝒃

⟩

2

<

||

𝒂

||

2

⁢

||

𝒃

||

2

,

which is the

Cauchy-Schwarz inequality

if

𝒂

and

𝒃

are linearly independent.

Title

proof of Cauchy-Schwarz inequality

Canonical name

ProofOfCauchySchwarzInequality

Date of creation

2013-03-22 12:34:42

Last modified on

2013-03-22 12:34:42

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

6

Author

mathwizard (128)

Entry type

Proof

Classification

msc 15A63

Generated on Fri Feb 9 18:30:17 2018 by

LaTeXML
proof of Cauchy-Schwarz inequality for real numbers

proof of Cauchy-Schwarz inequality for real numbers

The version of the

Cauchy-Schwartz inequality

we want to prove is

(

∑

k

=

1

n

a

k

⁢

b

k

)

2

≤

∑

k

=

1

n

a

k

2

⋅

∑

k

=

1

n

b

k

2

,

where the

a

k

and

b

k

are real numbers, with equality holding only in the
case of proportionality,

a

k

=

λ

⁢

b

k

for some real

λ

for all

k

.

The proof is by direct calculation:

∑

k

=

1

n

a

k

2

⋅

∑

k

=

1

n

b

k

2

-

(

∑

k

=

1

n

a

k

⁢

b

k

)

2

=

∑

k

,

l

=

1

n

a

k

2

⁢

b

l

2

-

a

k

⁢

b

k

⁢

a

l

⁢

b

l

=

∑

k

,

l

=

1

n

1

2

⁢

(

a

k

2

⁢

b

l

2

+

a

l

2

⁢

b

k

2

)

-

(

a

k

⁢

b

l

)

⁢

(

a

l

⁢

b

k

)

=

1

2

⁢

∑

k

,

l

=

1

n

(

a

k

⁢

b

l

)

2

-

2

⁢

(

a

k

⁢

b

l

)

⁢

(

a

l

⁢

b

k

)

+

(

a

l

⁢

b

k

)

2

=

1

2

⁢

∑

k

,

l

=

1

n

(

a

k

⁢

b

l

-

a

l

⁢

b

k

)

2

≥

0

.

The above identity implies that the Cauchy-Schwarz inequality holds.
Moreover, it is an equality only when

a

k

⁢

b

l

-

a

l

⁢

b

k

=

0

⟺

a

k

b

k

=

a

l

b

l

⁢

or

⁢

b

k

a

k

=

b

l

a

l

⁢

or

⁢

a

k

=

b

k

=

0

,

for all

k

and

l

. In other words, equality holds only when

a

k

=

λ

⁢

b

k

for all

k

for some real number

λ

.

Title

proof of Cauchy-Schwarz inequality for real numbers

Canonical name

ProofOfCauchySchwarzInequalityForRealNumbers

Date of creation

2013-03-22 14:56:38

Last modified on

2013-03-22 14:56:38

Owner

stitch (17269)

Last modified by

stitch (17269)

Numerical id

5

Author

stitch (17269)

Entry type

Proof

Classification

msc 15A63

Generated on Fri Feb 9 18:30:20 2018 by

LaTeXML
reflexive non-degenerate sesquilinear

reflexive non-degenerate sesquilinear

A

non-degenerate

sesquilinear form

b

:

V

×

V

→

k

is

reflexive

if for all

v

,

w

∈

V

, if

b

⁢

(

v

,

w

)

=

0

then

b

⁢

(

w

,

v

)

=

0

. This means

v

⟂

w

⁢

if and only if

⁢

w

⟂

v

.

It is rare to define

perpendicularity

for sesquilinear/bilinear maps which are not reflexive because it would require a version of left and right

perpendicular

. Thus a reflexive sesquilinear/bilinear map is usually synonymous with the

existence

of perpendicularity.

Title

reflexive non-degenerate sesquilinear

Canonical name

ReflexiveNondegenerateSesquilinear

Date of creation

2013-03-22 15:51:04

Last modified on

2013-03-22 15:51:04

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

10

Author

Algeboy (12884)

Entry type

Definition

Classification

msc 15A63

Synonym

reflexive non-degenerate bilinear

Synonym

reflexive sesquilinear

Synonym

reflexive bilinear

Related topic

SesquilinearFormsOverGeneralFields

Defines

Reflexive non-degenerate sesquilinear

Defines

Reflexive non-degenerate bilinear

Defines

Reflexive

Generated on Fri Feb 9 18:30:25 2018 by

LaTeXML
self-dual

self-dual

Definition.

Let

U

be a

finite-dimensional

inner-product space
over a field

𝕂

. Let

T

:

U

→

U

be an endomorphism,
and note that the

adjoint endomorphism

T

⋆

is also an endomorphism
of

U

. It is therefore possible to add, subtract, and compare

T

and

T

⋆

, and we are able to make the following

definitions

. An
endomorphism

T

is said to be

self-dual

(a.k.a.

self-adjoint

) if

T

=

T

⋆

.

By contrast, we say that the endomorphism is

anti self-dual

if

T

=

-

T

⋆

.

Exactly the same definitions can be made for an endomorphism of
a

complex vector space

with a

Hermitian inner product

.

Relation to the matrix transpose.

All of these definitions have
their counterparts in the matrix setting. Let

M

∈

Mat

n

,

n

(

𝕂

)

be the matrix of

T

relative to an

orthogonal
basis

of

U

. Then

T

is self-dual if and only if

M

is a

symmetric matrix

,
and anti self-dual if and only if

M

is a

skew-symmetric matrix

.

In the case of a Hermitian inner product we must replace the

transpose

with the

conjugate transpose

. Thus

T

is self dual if and only if

M

is a

Hermitian matrix

, i.e.

M

=

M

t

¯

.

It is anti self-dual if and only if

M

=

-

M

t

¯

.

Title

self-dual

Canonical

name

Selfdual

Date of creation

2013-03-22 12:29:40

Last modified on

2013-03-22 12:29:40

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

5

Author

rmilson (146)

Entry type

Definition

Classification

msc 15A63

Classification

msc 15A57

Classification

msc 15A04

Synonym

self-adjoint

Related topic

HermitianMatrix

Related topic

SymmetricMatrix

Related topic

SkewSymmetricMatrix

Defines

anti self-dual

Generated on Fri Feb 9 18:30:27 2018 by

LaTeXML
skew-symmetric bilinear form

skew-symmetric bilinear form

A

skew-symmetric

(or

antisymmetric

)

bilinear form

is a special case of a bilinear form

B

, namely one which is skew-symmetric in the two

coordinates

; that is,

B

⁢

(

x

,

y

)

=

-

B

⁢

(

y

,

x

)

for all vectors

x

and

y

. Note that this definition only makes sense if

B

is defined over two identical

vector spaces

, so we must require this in the formal definition:

a bilinear form

B

:

V

×

V

→

K

(

V

a vector
space over a field

K

) is called

skew-symmetric

iff

B

⁢

(

x

,

y

)

=

-

B

⁢

(

y

,

x

)

for all vectors

x

,

y

∈

V

.

Suppose that the

characteristic

of

K

is not

2

. Set

x

=

y

in the above equation. Then

B

⁢

(

x

,

x

)

=

-

B

⁢

(

x

,

x

)

for all vectors

x

∈

V

, which means that

2

⁢

B

⁢

(

x

,

x

)

=

0

, or

B

⁢

(

x

,

x

)

=

0

. Therefore,

B

is an

alternating form

.

If, however,

char

⁡

(

K

)

=

2

, then

B

⁢

(

x

,

y

)

=

-

B

⁢

(

y

,

x

)

=

B

⁢

(

y

,

x

)

;

B

is a

symmetric bilinear form

.

If

V

is

finite-dimensional

, then every bilinear form on

V

can be represented
by a matrix. In this case the following theorem applies:

A bilinear form is skew-symmetric iff its representing matrix is skew-symmetric.
(The fact that the representing matrix is skew-symmetric is independent of the
choice of representing matrix).

Title

skew-symmetric bilinear form

Canonical name

SkewsymmetricBilinearForm

Date of creation

2013-03-22 13:10:47

Last modified on

2013-03-22 13:10:47

Owner

sleske (997)

Last modified by

sleske (997)

Numerical id

9

Author

sleske (997)

Entry type

Definition

Classification

msc 15A63

Synonym

antisymmetric bilinear form

Synonym

anti-symmetric bilinear form

Related topic

AntiSymmetric

Related topic

SymmetricBilinearForm

Related topic

BilinearForm

Defines

skew symmetric

Defines

anti-symmetric

Defines

antisymmetric

Generated on Fri Feb 9 18:30:29 2018 by

LaTeXML
stufe of a field

stufe of a field

The

stufe

of a field

F

is the least number

n

such that

-

1

can be expressed as a sum of

n

squares:

-

1

=

a

1

2

+

⋯

+

a

n

2

,

where each

a

i

∈

F

. If no such an

n

exists, then we say that the

stufe

of

F

is

∞

.

Remarks

.

•

The word “stufe”, meaning “level” in German, is attributed to mathematician Albrecht Pfister.

•

A

theorem

of Pfister asserts that in a field

F

, if

-

1

can be expressed as a finite sum of squares, then the stufe of

F

is a power of

2

.

References

1

A. Pfister,

Zur Darstellung definiter Funktionen als Summe von Quadraten

, Inventiones Mathematicae (1967).

2

A. R. Rajwade,

Squares

, Cambridge University Press (1993).

Title

stufe of a field

Canonical

name

StufeOfAField

Date of creation

2013-03-22 15:06:01

Last modified on

2013-03-22 15:06:01

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 15A63

Classification

msc 12D15

Synonym

level of a field

Related topic

TheoremsOnSumsOfSquares

Defines

stufe

Generated on Fri Feb 9 18:30:31 2018 by

LaTeXML
unimodular

unimodular

Working over a

commutative ring

, a

bilinear form

on a module

V

is

unimodular

if it induces an

isomorphism

V

→

V

*

. Here

V

*

denotes the

dual module

of

V

.

Title

unimodular

Canonical name

Unimodular

Date of creation

2013-03-22 15:37:39

Last modified on

2013-03-22 15:37:39

Owner

whm22 (2009)

Last modified by

whm22 (2009)

Numerical id

10

Author

whm22 (2009)

Entry type

Definition

Classification

msc 15A63

Related topic

dualmodule

Related topic

DualModule

Related topic

DualSpace

Generated on Fri Feb 9 18:30:33 2018 by

LaTeXML
Clifford algebra

Clifford algebra

Let

V

be a

vector space

over a field

k

, and

Q

:

V

×

V

→

k

a

symmetric bilinear form

. Then the

Clifford algebra

Cliff

⁡

(

Q

,

V

)

is
the quotient of the

tensor algebra

𝒯

⁢

(

V

)

by the

relations

v

⊗

w

+

w

⊗

v

=

-

2

⁢

Q

⁢

(

v

,

w

)

∀

v

,

w

∈

V

.

Since the above relationship is not

homogeneous

in the usual

ℤ

-

grading

on

𝒯

⁢

(

V

)

,

Cliff

⁡

(

Q

,

V

)

does not inherit a

ℤ

-grading. However, by reducing mod 2, we also have a

ℤ

2

-grading on

𝒯

⁢

(

V

)

, and the relations above are homogeneous
with respect to this, so

Cliff

⁡

(

Q

,

V

)

has a natural

ℤ

2

-grading,
which makes it into a

superalgebra

.

In

addition

, we do have a

filtration

on

Cliff

⁡

(

Q

,

V

)

(making it a
filtered algebra), and the associated

graded algebra

Gr

⁡

Cliff

⁡

(

Q

,

V

)

is simply

Λ

*

⁢

V

, the

exterior algebra

of

V

. In
particular,

dim

⁡

Cliff

⁡

(

Q

,

V

)

=

dim

⁡

Λ

*

⁢

V

=

2

dim

⁡

V

.

The most commonly used Clifford algebra is the case

V

=

ℝ

n

, and

Q

is the standard

inner product

with

orthonormal basis

e

1

,

…

,

e

n

.
In this case, the

algebra

is generated by

e

1

,

…

,

e

n

and the

identity

of the algebra

1

, with the relations

e

i

2

=

-

1

e

i

⁢

e

j

=

-

e

j

e

i

(

i

≠

j

)

Trivially,

Cliff

⁡

(

ℝ

0

)

=

ℝ

, and it can be seen from the relations
above that

Cliff

⁡

(

ℝ

)

≅

ℂ

, the

complex numbers

, and

Cliff

⁡

(

ℝ

2

)

≅

ℍ

, the

quaternions

.

On the other ha
nd, for

V

=

ℂ

n

we get the particularly answer of

Cliff

⁡

(

ℂ

2

⁢

k

)

≅

M

2

k

⁢

(

ℂ

)

Cliff

⁡

(

ℂ

2

⁢

k

+

1

)

=

M

2

k

⁢

(

ℂ

)

⊕

𝐌

2

k

⁢

(

ℂ

)

.

Title

Clifford algebra

Canonical name

CliffordAlgebra

Date of creation

2013-03-22 13:18:05

Last modified on

2013-03-22 13:18:05

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

9

Author

rmilson (146)

Entry type

Definition

Classification

msc 15A66

Classification

msc 11E88

Generated on Fri Feb 9 18:30:35 2018 by

LaTeXML
antisymmetric mapping

antisymmetric mapping

Let

U

and

V

be a

vector spaces

over a field

K

. A bilinear mapping

B

:

U

×

U

→

V

is said to be

antisymmetric

if

B

⁢

(

u

,

u

)

=

0

(1)

for all

u

∈

U

.

If

B

is antisymmetric, then the

polarization

of the anti-symmetry

relation

gives the condition:

B

⁢

(

u

,

v

)

+

B

⁢

(

v

,

u

)

=

0

(2)

for all

u

,

v

∈

U

. If the

characteristic

of

K

is not 2, then
the two conditions are

equivalent

.

A multlinear

mapping

M

:

U

k

→

V

is said to be

totally antisymmetric

, or simply antisymmetric, if
for every

u

1

,

…

,

u

k

∈

U

such that

u

i

+

1

=

u

i

for some

i

=

1

,

…

,

k

-

1

we have

M

⁢

(

u

1

,

…

,

u

k

)

=

0

.

Proposition 1

Let

M

:

U

k

→

V

be a totally antisymmetric, multlinear
mapping, and let

π

be a

permutation

of

{

1

,

…

,

k

}

. Then,
for every

u

1

,

…

,

u

k

∈

U

we have

M

⁢

(

u

π

1

,

…

,

u

π

k

)

=

sgn

⁢

(

π

)

⁢

M

⁢

(

u

1

,

…

,

u

k

)

,

where

sgn

⁢

(

π

)

=

±

1

according to the parity of

π

.

Proof.

Let

u

1

,

…

,

u

k

∈

U

be given. multlinearity and anti-symmetry
imply that

0

=

M

⁢

(

u

1

+

u

2

,

u

1

+

u

2

,

u

3

,

…

,

u

k

)

=

M

⁢

(

u

1

,

u

2

,

u

3

,

…

,

u

k

)

+

M

⁢

(

u

2

,

u

1

,

u

3

,

…

,

u

k

)

Hence, the

proposition

is valid for

π

=

(

12

)

(see

cycle notation

).
Similarly, one can show that the proposition holds for all

transpositions

π

=

(

i

,

i

+

1

)

,

i

=

1

,

…

,

k

-

1

.

However, such transpositions
generate the group of permutations, and hence the proposition holds in
full generality.

Note.

The

determinant

is an excellent example of a totally
antisymmetric, multlinear mapping.

Title

antisymmetric mapping

Canonical

name

AntisymmetricMapping

Date of creation

2013-03-22 12:34:39

Last modified on

2013-03-22 12:34:39

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

10

Author

rmilson (146)

Entry type

Definition

Classification

msc 15A69

Classification

msc 15A63

Synonym

skew-symmetric

Synonym

anti-symmetric

Synonym

antisymmetric

Synonym

skew-symmetric mapping

Related topic

SkewSymmetricMatrix

Related topic

SymmetricBilinearForm

Related topic

ExteriorAlgebra

Generated on Fri Feb 9 18:30:43 2018 by

LaTeXML
coboundary definition of exterior derivative

coboundary definition of exterior derivative

Let

M

be a

smooth manifold

, and

•

let

C

∞

⁢

(

M

)

denote the

algebra

of

smooth functions

on

M

;

•

let

V

⁢

(

M

)

denote the Lie-algebra of smooth

vector fields

;

•

and let

Ω

k

⁢

(

M

)

denote the

vector space

of smooth,

differential

k

-forms.

Recall that a

differential form

α

∈

Ω

k

⁢

(

M

)

is a

multilinear

,

alternating

mapping

α

:

V

⁢

(

M

)

×

⋯

×

V

⁢

(

M

)

⁢

(k times)

→

C

∞

⁢

(

M

)

such that, in local coordinates,

α

looks like a multilinear combination of its vector field

arguments

. Thus, employing the

Einstein summation convention

and local
coordinates , we have

α

⁢

(

u

,

v

,

…

,

w

)

=

α

i

⁢

j

⁢

…

⁢

k

⁢

u

i

⁢

v

j

⁢

⋯

⁢

w

k

,

where

u

,

v

,

…

,

w

is a list of

k

vector fields. Recall also
that

C

∞

⁢

(

M

)

is a

V

⁢

(

M

)

module. The action is given by
a

directional derivative

, and takes the form

v

⁢

(

f

)

=

v

i

⁢

∂

i

⁡

f

,

v

∈

V

⁢

(

M

)

,

f

∈

C

∞

⁢

(

M

)

.

With these preliminaries out of the way, we have the following
description of the

exterior derivative

operator

d

:

Ω

k

⁢

(

M

)

→

Ω

k

+

1

⁢

(

M

)

. For

ω

∈

Ω

k

⁢

(

M

)

, we have

(

d

⁢

ω

)

⁢

(

v

0

,

v

1

,

…

,

v

k

)

=

∑

0

≤

i

≤

k

(

-

1

)

k

⁢

v

i

⁢

ω

⁢

(

…

,

v

^

i

,

…

)

+

(1)

+

∑

0

≤

i

<

j

≤

k

(

-

1

)

i

+

j

⁢

ω

⁢

(

[

v

i

,

v

j

]

,

…

,

v

^

i

,

…

⁢

v

^

j

,

…

)

,

where

v

^

i

indicates the omission of the argument

v

i

.

The above

expression

(

1

) of

d

⁢

ω

can be taken as
the

definition

of the exterior derivative. Letting the

v

i

arguments be

coordinate vector

fields, it is not hard to show that the above definition is equivalent to the
usual definition of

d

as a derivation of the

exterior algebra

of
differential forms, or the local coordinate definition of

d

. The
nice feature of (

1

) is that it is equivalent to the
definition of the

coboundary operator

for

Lie algebra cohomology

.
Thus, we see that

de Rham cohomology

, which is the

cohomology

of the

cochain complex

d

:

Ω

k

⁢

(

M

)

→

Ω

k

+

1

⁢

(

M

)

, is just
zeroth-order Lie algebra cohomology of

V

⁢

(

M

)

with

coefficients

in

C

∞

⁢

(

M

)

. The bit about “zeroth order” means that we are
considering cochains that are zeroth order

differential operators

of
their arguments — in other words, differential forms.

Title

coboundary definition of exterior derivative

Canonical

name

CoboundaryDefinitionOfExteriorDerivative

Date of creation

2013-03-22 15:38:06

Last modified on

2013-03-22 15:38:06

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

15

Author

rmilson (146)

Entry type

Definition

Classification

msc 15A69

Classification

msc 58A10

Related topic

LieAlgebraCohomology

Generated on Fri Feb 9 18:30:49 2018 by

LaTeXML
multi-linear

multi-linear

Let

V

1

,

V

2

,

…

,

V

n

,

W

be

vector spaces

over a field

K

. A
mapping

M

:

V

1

×

V

2

×

⋯

×

V

n

→

W

is
called

multi-linear

or

n

-linear, if

M

is linear in each of
its arguments.

Notes.

•

A

bilinear mapping

is another name for a

2

-

linear mapping

.

•

This definition generalizes in an obvious way to rings and
modules.

•

An excellent example of a multi-linear map is the

determinant

operation.

Title

multi-linear

Canonical name

Multilinear

Date of creation

2013-03-22 12:33:05

Last modified on

2013-03-22 12:33:05

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

10

Author

rmilson (146)

Entry type

Definition

Classification

msc 15A69

Synonym

multi-linearity

Synonym

multilinear

Synonym

multilinearity

Related topic

BilinearForm

Related topic

BilinearMap

Generated on Fri Feb 9 18:30:54 2018 by

LaTeXML
outer multiplication

outer multiplication

Note: the present entry employs the terminology and notation defined
and described in the entry on

tensor arrays

. To keep things
reasonably self

contained

we mention that the symbol

T

p

,

q

refers
to the

vector space

of type

(

p

,

q

)

tensor arrays, i.e. maps

I

p

×

I

q

→

𝕂

,

where

I

is some finite list of
index labels, and where

𝕂

is a field.

Let

p

1

,

p

2

,

q

1

,

q

2

be

natural numbers

.

Outer multiplication

is a

bilinear operation

T

p

1

,

q

1

×

T

p

2

,

q

2

→

T

p

1

+

p

2

,

q

1

+

q

2

that combines a type

(

p

1

,

q

1

)

tensor array

X

and a type

(

p

2

,

q

2

)

tensor array

Y

to
produce a type

(

p

1

+

p

2

,

q

1

+

q

2

)

tensor array

X

⁢

Y

(also
written as

X

⊗

Y

), defined by

(

X

⁢

Y

)

j

1

⁢

…

⁢

j

q

1

⁢

j

q

1

+

1

⁢

…

⁢

j

q

1

+

q

2

i

1

⁢

…

⁢

i

p

1

⁢

i

p

1

+

1

⁢

…

⁢

i

p

1

+

p

2

=

X

j

1

⁢

…

⁢

j

q

1

i

1

⁢

…

⁢

i

p

1

⁢

Y

j

q

1

+

1

⁢

…

⁢

j

q

1

+

q

2

i

p

1

+

1

⁢

…

⁢

i

p

1

+

p

2

Speaking informally, what is going on above is that we multiply
every value of the

X

array by every possible value of the

Y

array,
to create a new array,

X

⁢

Y

. Quite obviously then, the size of

X

⁢

Y

is
the size of

X

times the size of

Y

, and the index slots of the

product

X

⁢

Y

are just the union of the index slots of

X

and of

Y

.

Outer multiplication is a

non-commutative

,

associative

operation

. The
type

(

0

,

0

)

arrays are the scalars, i.e. elements of

𝕂

; they commute with everything. Thus, we can embed

𝕂

into
the

direct sum

⊕

p

,

q

∈

ℕ

T

p

,

q

,

and thereby endow the latter
with the

structure

of an

𝕂

-

algebra

1

1

We will not pursue this
line of thought here, because the topic of algebra structure is best
dealt with in the a more abstract

context

. The same comment applies
to the use of the

tensor product

sign

⊗

in denoting outer
multiplication. These topics are dealt with in the entry pertaining
to abstract

tensor algebra

.

.

By way of illustration we mention that the

outer product

of a

column
vector

, i.e. a type

(

1

,

0

)

array, and a

row vector

, i.e. a type

(

0

,

1

)

array, gives a matrix, i.e. a type

(

1

,

1

)

tensor array. For

instance

:

(

a

b

c

)

⊗

(

x

y

z

)

=

(

a

⁢

x

a

⁢

y

a

⁢

z

b

⁢

x

b

⁢

y

b

⁢

z

c

⁢

x

c

⁢

y

c

⁢

z

)

,

a

,

b

,

c

,

x

,

y

,

z

∈

𝕂

Title

outer multiplication

Canonical name

OuterMultiplication

Date of creation

2013-03-22 12:40:31

Last modified on

2013-03-22 12:40:31

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

4

Author

rmilson (146)

Entry type

Definition

Classification

msc 15A69

Related topic

TensorProductClassical

Related topic

TensorProduct

Generated on Fri Feb 9 18:30:58 2018 by

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second order tensor: symmetric and skew-symmetric parts

second order tensor: symmetric and skew-symmetric parts

We shall prove the following

theorem

on

existence

and uniqueness. (Here,
we assime that the ground field has

characteristic

different from 2.
This

hypothesis

is satisfied for the cases of greatest

interest

,
namely real and complex ground fields.)

Theorem 1

.

Every covariant and

contravariant tensor

of second rank may be expressed univocally as the sum of a

symmetric

and skew-symmetric tensor.

Proof.

Let us consider a contravariant tensor.

1. Existence.

Put

U

i

⁢

j

=

1

2

⁢

(

T

i

⁢

j

+

T

j

⁢

i

)

,

W

i

⁢

j

=

1

2

⁢

(

T

i

⁢

j

-

T

j

⁢

i

)

.

Then

U

i

⁢

j

=

U

j

⁢

i

is symmetric,

W

i

⁢

j

=

-

W

j

⁢

i

is

skew-symmetric

, and

T

i

⁢

j

=

U

i

⁢

j

+

W

i

⁢

j

.

2. Uniqueness.

Let us suppose that

T

i

⁢

j

admits the

decompositions

T

i

⁢

j

=

U

i

⁢

j

+

W

i

⁢

j

=

U

′

⁣

i

⁢

j

+

W

′

⁣

i

⁢

j

.

By taking the

transposes

T

j

⁢

i

=

U

j

⁢

i

+

W

j

⁢

i

=

U

′

⁣

j

⁢

i

+

W

′

⁣

j

⁢

i

,

we separate the symmetric and skew-symmetric parts in both

equations

and making use of their

symmetry

properties

, we have

U

i

⁢

j

-

U

′

⁣

i

⁢

j

=

W

′

⁣

i

⁢

j

-

W

i

⁢

j

=

U

j

⁢

i

-

U

′

⁣

j

⁢

i

=

W

′

⁣

j

⁢

i

-

W

j

⁢

i

=

W

i

⁢

j

-

W

′

⁣

i

⁢

j

=

U

′

⁣

i

⁢

j

-

U

i

⁢

j

=

-

(

U

i

⁢

j

-

U

′

⁣

i

⁢

j

)

=

0

,

which shows uniqueness of each part.

mutatis mutandis

for a

covariant tensor

T

i

⁢

j

.
∎

Title

second order

tensor: symmetric and skew-symmetric parts

Canonical

name

SecondOrderTensorSymmetricAndSkewsymmetricParts

Date of creation

2013-03-22 15:51:32

Last modified on

2013-03-22 15:51:32

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

18

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 15A69

Generated on Fri Feb 9 18:31:03 2018 by

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