aarjaneiro/planetmath huggingface dataset:
Householder transformation

Householder transformation

This entry describes the

Householder transformation

u

=

H

⁢

v

, the most frequently used algorithm for performing

QR decomposition

. The key object here is the

Householder matrix

H

, a

symmetric

and

orthogonal matrix

of the form

H

=

I

-

2

⁢

x

⁢

x

T

,

where

I

is the

identity matrix

and we have used any

normalized vector

x

with

||

x

||

2

2

=

x

T

⁢

x

=

1

.

The Householder transformation zeroes the last

m

-

1

elements of a

column vector

below the first element:

[

v

1

v

2

⋮

v

m

]

→

[

c

0

⋮

0

]

⁢

with

⁢

c

=

±

||

v

||

2

=

±

(

∑

i

=

1

m

v

i

2

)

1

/

2

One can verify that

x

=

f

⁢

[

v

1

-

c

v

2

⋮

v

m

]

⁢

with

⁢

f

=

1

2

⁢

c

⁢

(

c

-

v

1

)

fulfils

x

T

⁢

x

=

1

and that with

H

=

I

-

2

⁢

x

⁢

x

T

one obtains the vector

[

c

0

⋯

0

]

T

.

To perform the

decomposition

of the

m

×

n

matrix

A

=

Q

⁢

R

(with

m

≥

n

) we construct an

m

×

m

matrix

H

(

1

)

to change the

m

-

1

elements of the first column to zero. Similarly, an

m

-

1

×

m

-

1

matrix

G

(

2

)

will change the

m

-

2

elements of the second column to zero. With

G

(

2

)

we produce the

m

×

m

matrix

H

(

2

)

=

[

1

0

⋯

0

0

⋮

G

(

2

)

0

]

.

After

n

such

orthogonal transformations

(

n

-

1

times in the case that

m

=

n

), we let

R

=

H

(

n

)

⁢

⋯

⁢

H

(

2

)

⁢

H

(

1

)

⁢

A

.

R

is

upper triangular

and the orthogonal matrix

Q

becomes

Q

=

H

(

1

)

⁢

H

(

2

)

⁢

⋯

⁢

H

(

n

)

.

In practice the

H

(

i

)

are never explicitly computed.

References

•

Originally from The Data Analysis Briefbook
(

http://rkb.home.cern.ch/rkb/titleA.html

http://rkb.home.cern.ch/rkb/titleA.html)

Title

Householder transformation

Canonical name

HouseholderTransformation

Date of creation

2013-03-22 12:06:07

Last modified on

2013-03-22 12:06:07

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

10

Author

mathcam (2727)

Entry type

Algorithm

Classification

msc 15A57

Classification

msc 65F25

Synonym

Householder reflection

Synonym

Householder matrix

Related topic

GramSchmidtOrthogonalization

Generated on Fri Feb 9 18:29:07 2018 by

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M-matrix

M-matrix

A

Z-matrix

A

is called an

M-matrix

if it

satisfies

any one of
the following

equivalent

conditions.

1.

All

principal minors

of

A

are

positive

.

2.

The leading principal minors of

A

are positive.

3.

A

can be written in the form

A

=

k

⁢

I

-

B

, where

B

is a
non-negative matrix whose

spectral radius

is

strictly

less than

k

.

4.

All real

eigenvalues

of

A

are positive.

5.

The real part of any eigenvalue of

A

is positive.

6.

A

is

non-singular

and the

inverse

of

A

is non-negative.

7.

A

⁢

v

≥

0

implies

v

≥

0

.

8.

There exists a vector

v

with non-negative entries such
that

A

⁢

v

>

0

.

9.

A

+

D

is non-singular for every non-negative

diagonal matrix

D

.

10.

A

+

k

⁢

I

is non-singular for all

k

≥

0

.

11.

For each nonzero vector

v

,

v

i

⁢

(

A

⁢

v

)

i

>

0

for some

i

.

12.

There is a positive diagonal matrix

D

such that the matrix

D

⁢

A

+

A

T

⁢

D

is

positive definite

.

13.

A

can be factorized as

L

⁢

U

, where

L

is

lower
triangular

,

U

is upper triangular, and the

diagonal

entries of
both

L

and

U

are positive.

14.

The diagonal entries of

A

are positive and

A

⁢

D

is

strictly diagonally dominant

for some positive diagonal matrix

D

.

Reference:

M. Fiedler,

Special Matrices

and Their Applications in
Numerical Mathematics

, Martinus Nijhoff, Dordrecht, 1986.

R. A. Horn and C. R. Johnson,

Topics in Matrix Analysis

,
Cambridge University Press, Cambridge, 1991.

Title

M-matrix

Canonical

name

Mmatrix

Date of creation

2013-03-22 15:24:54

Last modified on

2013-03-22 15:24:54

Owner

kshum (5987)

Last modified by

kshum (5987)

Numerical id

7

Author

kshum (5987)

Entry type

Definition

Classification

msc 15A57

Generated on Fri Feb 9 18:29:10 2018 by

LaTeXML
Pauli matrices

Pauli matrices

The

Pauli matrices

are a set of three

Hermitian

,

unitary matrices

used by Wolfgang Pauli in his theory of quantum-mechanical spin. They are given by:

σ

1

=

(

0

1

1

0

)

σ

2

=

(

0

-

i

i

0

)

σ

3

=

(

1

0

0

-

1

)

They satisfy the following commutation and anticommutation identities:

[

σ

i

,

σ

j

]

=

2

⁢

i

⁢

ϵ

i

⁢

j

⁢

k

⁢

σ

k

⁢

where

ϵ

i

⁢

j

⁢

k

is the Levi-Civita symbol

{

σ

i

,

σ

j

}

=

2

⁢

𝐈

⁢

δ

i

⁢

j

⁢

where

𝐈

is the identity matrix and

δ

i

⁢

j

is the Kronecker delta

0.1

Delta notation

With the

identity matrix

I

, the Pauli matrices form a group. When combined in this way, they are often given the symbols

δ

i

, as follows:

δ

0

=

(

1

0

0

1

)

δ

1

=

(

0

1

1

0

)

δ

2

=

(

0

-

i

i

0

)

δ

3

=

(

1

0

0

-

1

)

This choice is useful when writing the

Dirac matrices

.

Title

Pauli matrices

Canonical name

PauliMatrices

Date of creation

2013-03-22 17:57:01

Last modified on

2013-03-22 17:57:01

Owner

invisiblerhino (19637)

Last modified by

invisiblerhino (19637)

Numerical id

9

Author

invisiblerhino (19637)

Entry type

Definition

Classification

msc 15A57

Synonym

sigma matrices

Related topic

Spinor

Related topic

SchrodingersWaveEquation

Related topic

UnitaryGroup

Related topic

HermitianMatrix

Related topic

DiracMatrices

Related topic

DiracEquation

Generated on Fri Feb 9 18:29:12 2018 by

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proof of determinant of the Vandermonde matrix

proof of determinant of the Vandermonde matrix

To begin, note that the

determinant

of the

n

×

n

Vandermonde
matrix

(which we shall denote as ‘

Δ

’) is a

homogeneous
polynomial

of order

n

⁢

(

n

-

1

)

/

2

because every term in the determinant
is, up to sign, the

product

of a zeroth power of some

variable

times the first
power of some other variable ,

…

, the

n

-

1

-st power of some
variable and

0

+

1

+

⋯

+

(

n

-

1

)

=

n

⁢

(

n

-

1

)

/

2

.

Next, note that if

a

i

=

a

j

with

i

≠

j

, then

Δ

=

0

because two columns of the matrix would be equal. Since

Δ

is a

polynomial

, this implies that

a

i

-

a

j

is a factor of

Δ

.
Hence,

Δ

=

C

⁢

∏

1

≤

i

<

j

≤

n

(

a

j

-

a

i

)

where C is some polynomial. However, since both

Δ

and the
product on the right hand side have the same degree,

C

must have
degree zero, i.e.

C

must be a constant. So all that remains is the
determine the value of this constant.

One way to determine this constant is to look at the

coefficient

of
the leading diagonal,

∏

n

(

a

n

)

n

-

1

. Since it equals 1 in both
the determinant and the product, we conclude that

C

=

1

, hence

Δ

=

∏

1

≤

i

<

j

≤

n

(

a

j

-

a

i

)

.

Title

proof of determinant of the Vandermonde matrix

Canonical name

ProofOfDeterminantOfTheVandermondeMatrix

Date of creation

2013-03-22 15:44:50

Last modified on

2013-03-22 15:44:50

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

10

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 15A57

Classification

msc 65F99

Classification

msc 65T50

Generated on Fri Feb 9 18:29:15 2018 by

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skew-Hermitian matrix

skew-Hermitian matrix

Definition

.

A

square matrix

A

with

complex

entries is

skew-Hermitian

, if

A

*

=

-

A

.

Here

A

∗

=

A

T

¯

,

A

T

is the

transpose

of

A

, and

A

¯

is
is the

complex conjugate

of the matrix

A

.

Properties.

1.

The trace of a

skew-Hermitian matrix

is

http://planetmath.org/node/2017

imaginary

.

2.

The

eigenvalues

of a skew-Hermitian matrix are

http://planetmath.org/node/2017

imaginary.

Proof.

Property

(1) follows directly from property (2) since the
trace is the sum of the eigenvalues. But one can also give a simple proof
as follows. Let

x

i

⁢

j

and

y

i

⁢

j

be the
real respectively

imaginary parts

of the elements in

A

.
Then the

diagonal

elements of

A

are of the
form

x

k

⁢

k

+

i

⁢

y

k

⁢

k

, and the diagonal elements in

A

∗

are of the form

-

x

k

⁢

k

+

i

⁢

y

k

⁢

k

. Hence

x

k

⁢

k

, i.e., the

real
part

for the diagonal elements in

A

must vanish, and
property (1) follows.
For property (2), suppose

A

is a skew-Hermitian matrix, and

x

an

eigenvector

corresponding to the eigenvalue

λ

, i.e.,

A

⁢

x

=

λ

⁢

x

.

(1)

Here,

x

is a complex

column vector

.
Since

x

is an eigenvector,

x

is not the

zero vector

, and

x

∗

⁢

x

>

0

. Without loss of generality we can assume

x

∗

⁢

x

=

1

.
Thus

λ

¯

=

x

∗

⁢

λ

¯

⁢

x

=

(

x

∗

⁢

λ

⁢

x

)

∗

=

(

x

∗

⁢

A

⁢

x

)

∗

=

x

∗

⁢

A

∗

⁢

x

=

x

∗

⁢

(

-

A

)

⁢

x

=

-

x

∗

⁢

λ

⁢

x

=

-

λ

.

Hence the eigenvalue

λ

corresponding
to

x

is

http://planetmath.org/node/2017

imaginary.

□

Title

skew-Hermitian matrix

Canonical

name

SkewHermitianMatrix

Date of creation

2013-03-22 13:36:14

Last modified on

2013-03-22 13:36:14

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

21

Author

matte (1858)

Entry type

Definition

Classification

msc 15A57

Synonym

anti-Hermitian matrix

Related topic

HermitianMatrix

Related topic

SymmetricMatrix

Related topic

SkewSymmetricMatrix

Generated on Fri Feb 9 18:29:17 2018 by

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theorem for normal triangular matrices

theorem for normal triangular matrices

Theorem 1

(

[

1

]

, pp. 82)
A

square matrix

is

diagonal

if and only if it is normal and triangular.

Proof.

If

A

is a

diagonal matrix

, then the

complex conjugate

A

∗

is also a diagonal matrix. Since arbitrary diagonal matrices
commute, it follows that

A

∗

⁢

A

=

A

⁢

A

∗

.
Thus
any diagonal matrix is a normal

triangular matrix

.

Next, suppose

A

=

(

a

i

⁢

j

)

is a normal

upper triangular matrix

.
Thus

a

i

⁢

j

=

0

for

i

>

j

, so for the diagonal

elements

in

A

∗

⁢

A

and

A

⁢

A

∗

, we obtain

(

A

∗

⁢

A

)

i

⁢

i

=

∑

k

=

1

i

|

a

k

⁢

i

|

2

,

(

A

⁢

A

∗

)

i

⁢

i

=

∑

k

=

i

n

|

a

i

⁢

k

|

2

.

For

i

=

1

, we have

|

a

11

|

2

=

|

a

11

|

2

+

|

a

12

|

2

+

⋯

+

|

a

1

⁢

n

|

2

.

It follows that the only non-zero entry on the first row of

A

is

a

11

.
Similarly, for

i

=

2

, we obtain

|

a

12

|

2

+

|

a

22

|

2

=

|

a

22

|

2

+

⋯

+

|

a

2

⁢

n

|

2

.

Since

a

12

=

0

, it follows that the only non-zero element on the
second row is

a

22

. Repeating this for all rows,
we see that

A

is a diagonal matrix. Thus any normal
upper triangular matrix is a diagonal matrix.

Suppose then that

A

is a normal

lower triangular matrix

.
Then it is not difficult to see that

A

∗

is a normal
upper triangular matrix. Thus, by the above,

A

∗

is a diagonal matrix,
whence also

A

is a diagonal matrix.

□

References

1

V.V. Prasolov,

Problems and Theorems in

Linear Algebra

,

American Mathematical Society

, 1994.

Title

theorem for normal triangular matrices

Canonical name

TheoremForNormalTriangularMatrices

Date of creation

2013-03-22 13:43:35

Last modified on

2013-03-22 13:43:35

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

12

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 15A57

Classification

msc 15-00

Related topic

NormalMatrix

Generated on Fri Feb 9 18:29:19 2018 by

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transpose

transpose

The

transpose

of a matrix

A

is the matrix formed by “flipping”

A

about the

diagonal

line from the upper left corner. It is usually denoted

A

t

, although sometimes it is written as

A

T

or

A

′

. So if

A

is an

m

×

n

matrix and

A

=

(

a

11

a

12

⋯

a

1

⁢

n

a

21

a

22

⋯

a

2

⁢

n

⋮

⋮

⋱

⋮

a

m

⁢

1

a

m

⁢

2

⋯

a

m

⁢

n

)

then

A

t

=

(

a

11

a

21

⋯

a

m

⁢

1

a

12

a

22

⋯

a

m

⁢

2

⋮

⋮

⋱

⋮

a

1

⁢

n

a

2

⁢

n

⋯

a

n

⁢

m

)

Note that the transpose of an

m

×

n

matrix is a

n

×

m

matrix.

Properties

Let

A

and

B

be

m

×

m

matrices,

C

and

D

be

m

×

n

matrices,

E

be an

n

×

k

matrix, and

c

be a constant. Let

x

and

y

be

column vectors

with

n

rows. Then

1.

(

C

t

)

t

=

C

2.

(

C

+

D

)

t

=

C

t

+

D

t

3.

(

c

⁢

D

)

t

=

c

⁢

D

t

4.

(

D

⁢

E

)

t

=

E

t

⁢

D

t

.

5.

(

A

⁢

B

)

t

=

B

t

⁢

A

t

.

6.

If

A

is

invertible

, then

(

A

t

)

-

1

=

(

A

-

1

)

t

7.

If

A

is real,

trace

⁡

(

A

t

⁢

A

)

≥

0

(where

trace

is the trace of a matrix).

8.

The transpose is a

linear mapping

from the

vector space

of matrices to itself. That is,

(

α

⁢

A

+

β

⁢

B

)

t

=

α

⁢

(

A

)

t

+

β

⁢

(

B

)

t

, for same-sized matrices

A

and

B

and scalars

α

and

β

.

The familiar vector

dot product

can also be defined using the matrix transpose. If

x

and

y

are column vectors with

n

rows each,

x

t

⁢

y

=

x

⋅

y

which implies

x

t

⁢

x

=

x

⋅

x

=

||

x

||

2

2

which is another way of defining the square of the

vector Euclidean norm

.

Title

transpose

Canonical

name

Transpose

Date of creation

2013-03-22 12:01:02

Last modified on

2013-03-22 12:01:02

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

12

Author

mathcam (2727)

Entry type

Definition

Classification

msc 15A57

Related topic

AdjointEndomorphism

Related topic

HermitianConjugate

Related topic

FrobeniusMatrixNorm

Related topic

ConjugateTranspose

Related topic

TransposeOperator

Related topic

VectorizationOfMatrix

Generated on Fri Feb 9 18:29:22 2018 by

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Z-matrix

Z-matrix

A

square matrix

is called a

Z

-matrix

if all

off-diagonal entries

are less than or equal to zero.

Z-matrix

arises naturally in solving

Dirichlet problem

numerically, and linear modeling of economy.

Reference:

M. Fiedler,

Special matrices

and their applications in
numerical mathematics,

1986, Martinus Nijhoff Publishers,
Dordrecht.

R. A. Horn and C. R. Johnson,

Topics in matrix analysis,

1991, Cambridge University Press, United Kingdom.

R. B. Bapat and T. E. S. Raghavan,

Nonnegative matrices

and
applications,

1997, Cambridge University Press, United Kingdom.

Title

Z-matrix

Canonical name

Zmatrix

Date of creation

2013-03-22 15:24:51

Last modified on

2013-03-22 15:24:51

Owner

kshum (5987)

Last modified by

kshum (5987)

Numerical id

11

Author

kshum (5987)

Entry type

Definition

Classification

msc 15A57

Generated on Fri Feb 9 18:29:24 2018 by

LaTeXML
Frobenius product

Frobenius product

If

A

=

(

a

i

⁢

j

)

and

B

=

(

b

i

⁢

j

)

are real

m

×

n

matrices, their

Frobenius product

is defined as

⟨

A

,

B

⟩

F

:=

∑

i

,

j

a

i

⁢

j

⁢

b

i

⁢

j

.

It is easily seen that

⟨

A

,

B

⟩

F

is equal to the trace of the matrix

A

⊺

⁢

B

and

A

⁢

B

⊺

, and that the Frobenius product is an

inner product

of the

vector space

formed by the

m

×

n

matrices; it the

Frobenius norm

of this vector space.

Title

Frobenius product

Canonical name

FrobeniusProduct

Date of creation

2013-03-22 18:11:34

Last modified on

2013-03-22 18:11:34

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Definition

Classification

msc 15A60

Classification

msc 15A63

Synonym

Frobenius inner product

Related topic

NormedVectorSpace

Related topic

FrobeniusMatrixNorm

Related topic

Product

Defines

Frobenius norm

Generated on Fri Feb 9 18:29:26 2018 by

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Hölder inequality

Hölder inequality

The

Hölder

inequality

concerns

vector p-norms

: given

1

≤

p

,

q

≤

∞

,

If

⁢

1

p

+

1

q

=

1

⁢

then

⁢

|

x

T

⁢

y

|

≤

||

x

||

p

⁢

||

y

||

q

An important instance of a Hölder inequality is the

Cauchy-Schwarz inequality

.

There is a version of this result for the

L

p

spaces (

http://planetmath.org/LpSpace

).
If a function

f

is in

L

p

⁢

(

X

)

, then the

L

p

-norm of

f

is denoted

||

f

||

p

.
Given a

measure space

(

X

,

𝔅

,

μ

)

, if

f

is in

L

p

⁢

(

X

)

and

g

is in

L

q

⁢

(

X

)

(with

1

/

p

+

1

/

q

=

1

), then
the Hölder inequality becomes

∥

f

⁢

g

∥

1

=

∫

X

|

f

⁢

g

|

⁢

d

μ

≤

(

∫

X

|

f

|

p

⁢

d

μ

)

1

p

⁢

(

∫

X

|

g

|

q

⁢

d

μ

)

1

q

=

∥

f

∥

p

⁢

∥

g

∥

q

Title

Hölder inequality

Canonical name

HolderInequality

Date of creation

2013-03-22 11:43:06

Last modified on

2013-03-22 11:43:06

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

27

Author

PrimeFan (13766)

Entry type

Theorem

Classification

msc 15A60

Classification

msc 55-XX

Classification

msc 46E30

Classification

msc 42B10

Classification

msc 42B05

Synonym

Holder inequality

Synonym

Hoelder inequality

Related topic

VectorPnorm

Related topic

CauchySchwartzInequality

Related topic

CauchySchwarzInequality

Related topic

ProofOfMinkowskiInequality

Related topic

ConjugateIndex

Related topic

BoundedLinearFunctionalsOnLpmu

Related topic

ConvolutionsOfComplexFunctionsOnLocallyCompactGroups

Related topic

LpNormIsDualToLq

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

LaTeXML
matrix p-norm

matrix p-norm

A class of

matrix norms

, denoted

∥

⋅

∥

p

, is defined as

∥

A

∥

p

=

sup

x

≠

0

⁡

∥

A

⁢

x

∥

p

∥

x

∥

p

x

∈

ℝ

n

,

A

∈

ℝ

m

×

n

.

The matrix

p

-norms are defined in terms of the

vector

p

-norms (

http://planetmath.org/VectorPNorm

)

.

An alternate

definition

is

∥

A

∥

p

=

max

∥

x

∥

p

=

1

⁢

∥

A

⁢

x

∥

p

.

As with vector

p

-norms, the most important are the 1, 2, and

∞

norms.
The 1 and

∞

norms are very easy to calculate for an arbitrary matrix:

∥

A

∥

1

=

max

1

≤

j

≤

n

⁢

∑

i

=

1

m

|

a

i

⁢

j

|

∥

A

∥

∞

=

max

1

≤

i

≤

m

⁢

∑

j

=

1

n

|

a

i

⁢

j

|

.

It directly follows from this that

∥

A

∥

1

=

∥

A

T

∥

∞

.

The calculation of the

2

-norm is more complicated. However, it can be shown that
the 2-norm of

A

is the square root of the largest

eigenvalue

of

A

T

⁢

A

.
There are also various

inequalities

that allow one to make

estimates

on the value
of

∥

A

∥

2

:

1

n

⁢

∥

A

∥

∞

≤

∥

A

∥

2

≤

m

⁢

∥

A

∥

∞

.

1

m

⁢

∥

A

∥

1

≤

∥

A

∥

2

≤

n

⁢

∥

A

∥

1

.

∥

A

∥

2

2

≤

∥

A

∥

∞

⋅

∥

A

∥

1

.

∥

A

∥

2

≤

∥

A

∥

F

≤

n

⁢

∥

A

∥

2

.

(

∥

A

∥

F

is the

Frobenius matrix norm

)

Title

matrix p-norm

Canonical

name

MatrixPnorm

Date of creation

2013-03-22 11:43:22

Last modified on

2013-03-22 11:43:22

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

20

Author

mathcam (2727)

Entry type

Definition

Classification

msc 15A60

Classification

msc 00A69

Related topic

MatrixNorm

Related topic

VectorNorm

Related topic

FrobeniusMatrixNorm

Generated on Fri Feb 9 18:29:31 2018 by

LaTeXML
proof of Hölder inequality

proof of Hölder inequality

First we prove the more general form (in

measure spaces

).

Let

(

X

,

μ

)

be a measure space and let

f

∈

L

p

⁢

(

X

)

,

g

∈

L

q

⁢

(

X

)

where

p

,

q

∈

[

1

,

+

∞

]

and

1

p

+

1

q

=

1

.

The case

p

=

1

and

q

=

∞

is obvious since

|

f

⁢

(

x

)

⁢

g

⁢

(

x

)

|

≤

∥

g

∥

L

∞

⁢

|

f

⁢

(

x

)

|

.

Also if

f

=

0

or

g

=

0

the result is obvious.
Otherwise notice that (applying

http://planetmath.org/node/YoungInequality

Young inequality

) we have

∥

f

⁢

g

∥

1

∥

f

∥

p

⋅

∥

g

∥

q

=

∫

X

|

f

|

∥

f

∥

p

⋅

|

g

|

∥

g

∥

q

⁢

𝑑

μ

≤

1

p

⁢

∫

X

(

|

f

|

∥

f

∥

p

)

p

⁢

𝑑

μ

+

1

q

⁢

∫

X

(

|

g

|

∥

g

∥

q

)

q

⁢

𝑑

μ

=

1

p

+

1

q

=

1

hence the desired

inequality

holds

∫

X

|

f

⁢

g

|

=

∥

f

⁢

g

∥

1

≤

∥

f

∥

p

⋅

∥

g

∥

q

=

(

∫

X

|

f

|

p

)

1

p

⁢

(

∫

X

|

g

|

q

)

1

q

.

If

x

and

y

are vectors in

ℝ

n

or vectors in

ℓ

p

and

ℓ

q

-spaces we can specialize the previous result by choosing

μ

to be the

counting measure

on

ℕ

.

In this case the proof can also be rewritten, without using

measure theory

,
as follows.
If we define

∥

x

∥

p

=

(

∑

k

|

x

k

|

p

)

1

p

we have

|

∑

k

x

k

⁢

y

k

|

∥

x

∥

p

⋅

∥

y

∥

q

≤

∑

k

|

x

k

|

⁢

|

y

k

|

∥

x

∥

p

⋅

∥

y

∥

q

=

∑

k

|

x

k

|

∥

x

∥

p

⁢

|

y

k

|

∥

y

∥

q

≤

1

p

⁢

∑

k

|

x

k

|

p

∥

x

∥

p

p

+

1

q

⁢

∑

k

|

y

k

|

q

∥

y

∥

q

q

=

1

p

+

1

q

=

1

.

Title

proof of Hölder inequality

Canonical name

ProofOfHolderInequality

Date of creation

2013-03-22 13:31:16

Last modified on

2013-03-22 13:31:16

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

10

Author

paolini (1187)

Entry type

Proof

Classification

msc 15A60

Classification

msc 46E30

Synonym

proof of Hölder inequality

Synonym

proof of Holder’s inequality

Generated on Fri Feb 9 18:29:34 2018 by

LaTeXML
self consistent matrix norm

self consistent matrix norm

A

matrix norm

N

is said to be

self

consistent

if

N

⁢

(

𝐀𝐁

)

≤

N

⁢

(

𝐀

)

⋅

N

⁢

(

𝐁

)

for all pairs of matrices

𝐀

and

𝐁

such that

𝐀𝐁

is defined.

Title

self consistent matrix norm

Canonical

name

SelfConsistentMatrixNorm

Date of creation

2013-03-22 13:39:22

Last modified on

2013-03-22 13:39:22

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

10

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 15A60

Related topic

GelfandSpectralRadiusTheorem

Defines

self consistent norm

Defines

self-consistent matrix norm

Defines

self-consistent norm

Defines

self-consistent

Defines

self consistent

Generated on Fri Feb 9 18:29:36 2018 by

LaTeXML