aarjaneiro/planetmath huggingface dataset:
generalized Vandermonde matrix

generalized Vandermonde matrix

A

generalized Vandermonde matrix

is a

square matrix

with entries in

ℝ

of the
form:

(

a

i

b

j

)

i

,

j

=

1

n

=

(

a

1

b

1

a

1

b

2

⋯

a

1

b

n

a

2

b

1

a

2

b

2

⋯

a

2

b

n

⋮

⋮

⋮

a

n

b

1

a

n

b

2

⋯

a

n

b

n

)

with real numbers

0

<

a

1

<

a

2

<

⋯

<

a

n

and

b

1

<

b

2

<

⋯

<

b

n

.

Title

generalized Vandermonde matrix

Canonical name

GeneralizedVandermondeMatrix

Date of creation

2013-03-22 17:23:30

Last modified on

2013-03-22 17:23:30

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

5

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 15A48

Synonym

generalised Vandermonde matrix

Generated on Fri Feb 9 18:28:17 2018 by

LaTeXML
generalized Vandermonde matrix is totally positive

generalized Vandermonde matrix is totally positive

A

generalized Vandermonde matrix

(

http://planetmath.org/GeneralizedVandermondeMatrix

) is

totally positive

.

Title

generalized Vandermonde matrix is totally positive

Canonical name

GeneralizedVandermondeMatrixIsTotallyPositive

Date of creation

2013-03-22 14:34:01

Last modified on

2013-03-22 14:34:01

Owner

GeraW (6138)

Last modified by

GeraW (6138)

Numerical id

14

Author

GeraW (6138)

Entry type

Theorem

Classification

msc 15A48

Generated on Fri Feb 9 18:28:19 2018 by

LaTeXML
negative definite

negative definite

Let

A

be an

n

×

n

symmetric

real

square matrix

. If for any non-zero vector

x

we have

x

t

⁢

A

⁢

x

<

0

,

we call

A

a

negative definite

matrix.

Title

negative definite

Canonical name

NegativeDefinite

Date of creation

2013-03-22 12:20:06

Last modified on

2013-03-22 12:20:06

Owner

drini (3)

Last modified by

drini (3)

Numerical id

4

Author

drini (3)

Entry type

Definition

Classification

msc 15A48

Related topic

PositiveDefinite

Related topic

PositiveSemidefinite

Related topic

QuadraticForm

Generated on Fri Feb 9 18:28:21 2018 by

LaTeXML
negative semidefinite

negative semidefinite

Let

A

be an

n

×

n

symmetric

real

square matrix

. If for any non-zero vector

x

we have

x

t

⁢

A

⁢

x

≤

0

,

we call

A

a

negative semidefinite

matrix.

Title

negative semidefinite

Canonical name

NegativeSemidefinite

Date of creation

2013-03-22 12:20:14

Last modified on

2013-03-22 12:20:14

Owner

drini (3)

Last modified by

drini (3)

Numerical id

4

Author

drini (3)

Entry type

Definition

Classification

msc 15A48

Generated on Fri Feb 9 18:28:22 2018 by

LaTeXML
positive definite

positive definite

Introduction

The definiteness of a matrix is an important

property

that has use in many areas of mathematics and physics.
Below are some examples:

1.

In optimizing problems, the definiteness of the
Hessian matrix determines the quality of an extremal value.
The full details can be found on
this page (

http://planetmath.org/RelationsBetweenHessianMatrixAndLocalExtrema

).

Definition

[

1

]

Suppose

A

is an

n

×

n

square

Hermitian matrix

.
If, for any non-zero vector

x

, we have that

x

∗

⁢

A

⁢

x

>

0

,

then

A

a

positive definite

matrix. (Here

x

∗

=

x

¯

t

,
where

x

¯

is the

complex conjugate

of

x

, and

x

t

is
the

transpose

of

x

.)

One can show that a Hermitian matrix is positive definite if
and only if all its

eigenvalues

are

positive

[

1

]

.
Thus the

determinant

of a

positive definite matrix

is positive, and
a positive definite matrix is always

invertible

.
The

Cholesky decomposition

provides an economical method for
solving

linear equations

involving a positive definite matrix.
Further conditions and properties for positive definite matrices
are given in

[

2

]

.

References

1

M. C. Pease,

Methods of Matrix Algebra

,
Academic Press, 1965

2

C.R. Johnson,

Positive definite matrices

,
American Mathematical Monthly, Vol. 77, Issue 3 (March 1970) 259-264.

Title

positive definite

Canonical

name

PositiveDefinite

Date of creation

2013-03-22 12:20:03

Last modified on

2013-03-22 12:20:03

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

10

Author

matte (1858)

Entry type

Definition

Classification

msc 15A48

Related topic

PositiveSemidefinite

Related topic

NegativeDefinite

Related topic

QuadraticForm

Related topic

EuclideanVectorSpace

Related topic

EuclideanVectorSpace2

Generated on Fri Feb 9 18:28:24 2018 by

LaTeXML
positive semidefinite

positive semidefinite

Let

A

be an

n

×

n

symmetric

real

square matrix

. If for any non-zero vector

x

we have

x

t

⁢

A

⁢

x

≥

0

,

we call

A

a

positive semidefinite

matrix.

Title

positive semidefinite

Canonical name

PositiveSemidefinite

Date of creation

2013-03-22 12:20:10

Last modified on

2013-03-22 12:20:10

Owner

drini (3)

Last modified by

drini (3)

Numerical id

5

Author

drini (3)

Entry type

Definition

Classification

msc 15A48

Related topic

PositiveDefinite

Related topic

NegativeDefinite

Generated on Fri Feb 9 18:28:26 2018 by

LaTeXML
reducible matrix

reducible matrix

An

n

×

n

matrix

A

is said to be a

reducible matrix

if and only if for some

permutation matrix

P

, the matrix

P

T

⁢

A

⁢

P

is block

upper triangular

.
If a

square matrix

is not

reducible

, it is said to be an

irreducible matrix

.

The following conditions on an

n

×

n

matrix

A

are

equivalent

.

1.

A

is an irreducible matrix.

2.

The

digraph

associated to

A

is

strongly connected

.

3.

For each

i

and

j

, there exists some

k

such that

(

A

k

)

i

⁢

j

>

0

.

4.

For any

partition

J

⊔

K

of the

index set

{

1

,

2

,

…

,

n

}

, there exist

j

∈

J

and

k

∈

K

such that

a

j

⁢

k

≠

0

.

For certain applications, irreducible matrices are more useful than reducible matrices. In particular, the

Perron-Frobenius theorem

gives more information about the spectra of irreducible matrices than of reducible matrices.

Title

reducible matrix

Canonical

name

ReducibleMatrix

Date of creation

2013-03-22 13:18:20

Last modified on

2013-03-22 13:18:20

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

11

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 15A48

Defines

irreducible matrix

Generated on Fri Feb 9 18:28:28 2018 by

LaTeXML
square root of positive definite matrix

square root of positive definite matrix

Suppose

M

is a

positive definite

Hermitian matrix

. Then

M

has a

diagonalization

M

=

P

*

⁢

diag

⁡

(

λ

1

,

…

,

λ

n

)

⁢

P

where

P

is a

unitary matrix

and

λ

1

,

…

,

λ

n

are the

eigenvalues

of

M

, which are all

positive

.

We can now define the

square root

of

M

as the matrix

M

1

/

2

=

P

*

⁢

diag

⁡

(

λ

1

,

…

,

λ

n

)

⁢

P

.

The following properties are clear

1.

M

1

/

2

⁢

M

1

/

2

=

M

,

2.

M

1

/

2

is Hermitian and positive definite.

3.

M

1

/

2

and

M

commute

4.

(

M

1

/

2

)

T

=

(

M

T

)

1

/

2

.

5.

(

M

1

/

2

)

-

1

=

(

M

-

1

)

1

/

2

, so one can write

M

-

1

/

2

6.

If the eigenvalues of

M

are

(

λ

1

,

…

,

λ

n

)

, then
the eigenvalues of

M

1

/

2

are

(

λ

1

,

…

,

λ

n

)

.

Title

square root of positive definite matrix

Canonical name

SquareRootOfPositiveDefiniteMatrix

Date of creation

2013-03-22 15:16:42

Last modified on

2013-03-22 15:16:42

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

12

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 15A48

Related topic

CholeskyDecomposition

Generated on Fri Feb 9 18:28:30 2018 by

LaTeXML
totally positive matrix

totally positive matrix

An

n

×

n

matrix over

ℝ

is said to be
if the

determinant

of every square

submatrix

is positive. Hence, the determinant
and every

element of

the matrix are positive.

Title

totally positive matrix

Canonical name

TotallyPositiveMatrix

Date of creation

2013-03-22 17:23:28

Last modified on

2013-03-22 17:23:28

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 15A48

Defines

totally positive

Generated on Fri Feb 9 18:28:32 2018 by

LaTeXML
Birkhoff-von Neumann theorem

Birkhoff-von Neumann theorem

An

n

×

n

matrix over

ℜ

is

doubly stochastic

if and only if it is a

convex combination

of

permutation matrices

.

Title

Birkhoff-von Neumann theorem

Canonical name

BirkhoffvonNeumannTheorem

Date of creation

2013-03-22 13:10:00

Last modified on

2013-03-22 13:10:00

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

6

Author

Andrea Ambrosio (7332)

Entry type

Theorem

Classification

msc 15A51

Related topic

CharacterizationsOfMajorization

Generated on Fri Feb 9 18:28:34 2018 by

LaTeXML
Google matrix

Google matrix

Google’s PageRank

algorithm

uses a particular

stochastic matrix

called the

Google matrix

.
The purpose of the PageRank algorithm is to compute a stationary vector of the Google matrix.
The stationary vector is then used to provide a ranking of the pages on the internet.

A

directed graph

D

is constructed whose vertices correspond to web pages and a directed
arc from vertex

i

to vertex

j

exists if and only if page

i

has a link out to
page

j

.
Then a stochastic matrix

A

=

(

a

i

⁢

j

)

is constructed from

D

: for each

i

,

j

set

a

i

⁢

j

=

1

/

d

⁢

(

i

)

if the

outdegree

of vertex

i

,

d

⁢

(

i

)

is

positive

and there is an arc from

i

to

j

in

D

.
Set

a

i

⁢

j

=

0

if

d

⁢

(

i

)

>

0

but there is no arc from

i

to

j

in

D

.

Set

a

i

⁢

j

=

1

/

n

if

d

⁢

(

i

)

=

0

, where

n

is the order of the matrix.

Having defined

A

choose a positive

row vector

v

T

such that

v

T

⁢

𝟏

=

1

where

1

is a vector of all ones.
Finally, choose a constant

c

∈

(

0

,

1

)

.
The

Google matrix

G

is

G

=

c

⁢

A

+

(

1

-

c

)

⁢

𝟏

⁢

v

T

.

Clearly,

G

is stochastic. For the actual matrix that Google uses

c

is about .85.

Title

Google matrix

Canonical

name

GoogleMatrix

Date of creation

2013-03-22 17:01:05

Last modified on

2013-03-22 17:01:05

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

6

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 15A51

Generated on Fri Feb 9 18:28:36 2018 by

LaTeXML
primitive matrix

primitive matrix

A nonnegative

square matrix

A

=

(

a

i

⁢

j

)

is said to be a if there exists

k

such that

A

k

≫

0

, i.e., if there exists

k

such that for all

i

,

j

, the

(

i

,

j

)

entry of

A

k

is

positive

.

A sufficient condition for a matrix to be a

primitive matrix

is for the matrix to
be a nonnegative,

irreducible matrix

with a

positive element

on the main diagonal.

Title

primitive matrix

Canonical name

PrimitiveMatrix

Date of creation

2013-03-22 13:18:18

Last modified on

2013-03-22 13:18:18

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

12

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 15A51

Classification

msc 15A48

Generated on Fri Feb 9 18:28:38 2018 by

LaTeXML
there are no non-square doubly stochastic matrices

there are no non-square doubly stochastic matrices

Suppose

A

=

(

a

i

⁢

j

)

is a

n

×

m

matrix with nonnegative entries such
that

∑

j

=

1

m

a

i

⁢

j

=

1

,

i

=

1

,

…

,

n

,

(1)

∑

i

=

1

n

a

i

⁢

j

=

1

,

j

=

1

,

…

,

m

.

(2)

Then

n

=

m

.

This is seen by summing equation
(

1

) over

i

=

1

,

…

,

n

and
equation (

2

) over

j

=

1

,

…

,

m

. Then

∑

i

=

1

n

∑

j

=

1

m

a

i

⁢

j

=

n

,

∑

i

=

1

n

∑

j

=

1

m

a

i

⁢

j

=

m

,

and since the right hand sides coincide, it follows that

n

=

m

.

Title

there are no non-square

doubly stochastic matrices

Canonical name

ThereAreNoNonsquareDoublyStochasticMatrices

Date of creation

2013-03-22 15:11:00

Last modified on

2013-03-22 15:11:00

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

8

Author

matte (1858)

Entry type

Result

Classification

msc 15A51

Classification

msc 60G99

Generated on Fri Feb 9 18:28:45 2018 by

LaTeXML
Cauchy matrix

Cauchy matrix

Let

x

1

,

x

2

,

…

,

x

m

, and

y

1

,

y

2

⁢

…

,

y

n

be elements in a field

F

, satisfying the that

1.

x

1

,

…

,

x

m

are distinct,

2.

y

1

,

…

,

y

n

are distinct, and

3.

x

i

+

y

j

≠

0

for

1

≤

i

≤

m

,

1

≤

j

≤

n

.

The matrix

[

1

x

1

+

y

1

1

x

1

+

y

2

⋯

1

x

1

+

y

n

1

x

2

+

y

1

1

x

2

+

y

2

⋯

1

x

2

+

y

n

⋮

⋮

⋱

⋮

1

x

m

+

y

1

1

x

m

+

y

2

⋯

1

x

m

+

y

n

]

is called a

Cauchy matrix

over

F

.

The

determinant

of a square Cauchy matrix is

∏

i

<

j

(

x

i

-

x

j

)

⁢

(

y

i

-

y

j

)

∏

i

⁢

j

(

x

i

+

y

j

)

Since

x

i

’s are distinct and

y

j

’s are distinct by

definition

, a square Cauchy matrix is non-singular. Any

submatrix

of a rectangular Cauchy matrix has full rank.

Title

Cauchy matrix

Canonical

name

CauchyMatrix

Date of creation

2013-03-22 14:30:43

Last modified on

2013-03-22 14:30:43

Owner

kshum (5987)

Last modified by

kshum (5987)

Numerical id

9

Author

kshum (5987)

Entry type

Definition

Classification

msc 15A57

Defines

Cauchy matrices

Generated on Fri Feb 9 18:28:48 2018 by

LaTeXML
condition of orthogonality

condition of orthogonality

Let two

straight lines

of the

x

⁢

y

-plane have the slopes

m

1

and

m

2

.  The lines are at

right angles

to each other iff

m

1

and

m

2

are the

opposite

inverses

of each other, i.e. iff (

http://planetmath.org/

Biconditional

)

m

1

⁢

m

2

=

-

1

.

Example.

The lines

y

=

(

1

+

2

)

⁢

x

and

y

=

(

1

-

2

)

⁢

x

are at right angles to each other.

Title

condition of orthogonality

Canonical name

ConditionOfOrthogonality

Date of creation

2013-03-22 14:48:05

Last modified on

2013-03-22 14:48:05

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

14

Author

pahio (2872)

Entry type

Result

Classification

msc 15A57

Classification

msc 51F20

Synonym

condition of perpendicularity

Related topic

OrthogonalCurves

Related topic

InverseNumber

Related topic

OppositeNumber

Related topic

NormalLine

Related topic

AngleBetweenTwoLines

Related topic

PerpendicularityInEuclideanPlane

Related topic

Evolute2

Related topic

ExampleOfFindingCatacaustic

Defines

opposite inverse

Generated on Fri Feb 9 18:28:50 2018 by

LaTeXML
determinant of the Vandermonde matrix

determinant of the Vandermonde matrix

Let

R

be a commutative ring with identity, and fix elements

a

1

,

a

2

,

…

,

a

n

∈

R

. The

determinant

of the

Vandermonde matrix

(

(

a

i

)

j

-

1

)

i

,

j

=

1

n

is equal to the product

∏

1

≤

i

<

j

≤

n

(

a

j

-

a

i

)

.

Title

determinant of the Vandermonde matrix

Canonical name

DeterminantOfTheVandermondeMatrix

Date of creation

2013-03-22 14:33:52

Last modified on

2013-03-22 14:33:52

Owner

GeraW (6138)

Last modified by

GeraW (6138)

Numerical id

10

Author

GeraW (6138)

Entry type

Result

Classification

msc 15A57

Classification

msc 65F99

Classification

msc 65T50

Synonym

Vandermonde determinant

Generated on Fri Feb 9 18:28:52 2018 by

LaTeXML
eigenvalues of a Hermitian matrix are real

eigenvalues of a Hermitian matrix are real

The

eigenvalues

of a

Hermitian

(or

self-adjoint

) matrix are real.

Proof.

Suppose

λ

is an eigenvalue of the

self-adjoint matrix

A

with
non-zero

eigenvector

v

. Then

A

⁢

v

=

λ

⁢

v

.

λ

∗

⁢

v

H

⁢

v

=

(

λ

⁢

v

)

H

⁢

v

=

(

A

⁢

v

)

H

⁢

v

=

v

H

⁢

A

H

⁢

v

=

v

H

⁢

A

⁢

v

=

v

H

⁢

λ

⁢

v

=

λ

⁢

v

H

⁢

v

Since

v

is non-zero by

assumption

,

v

H

⁢

v

is non-zero as well and so

λ

*

=

λ

, meaning that

λ

is real.
∎

Title

eigenvalues of a

Hermitian matrix

are real

Canonical name

EigenvaluesOfAHermitianMatrixAreReal

Date of creation

2013-03-22 14:23:09

Last modified on

2013-03-22 14:23:09

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

8

Author

Andrea Ambrosio (7332)

Entry type

Theorem

Classification

msc 15A57

Generated on Fri Feb 9 18:28:54 2018 by

LaTeXML
fully indecomposable matrix

fully indecomposable matrix

An

n

×

n

matrix

A

that contains an

s

×

(

n

-

s

)

zero

submatrix

for some

positive integer

s

is said to be

partly decomposable

. If no such submatrix exists then

A

is said to be

it fully indecomposable

.
By convention, a

1

×

1

matrix is fully indecomposable if it is nonzero.

A

is

nearly decomposable

if it fully indecomposable but whenever a nonzero entry is changed to 0 the resulting matrix is partly decomposable.

Title

fully indecomposable matrix

Canonical name

FullyIndecomposableMatrix

Date of creation

2013-03-22 15:58:56

Last modified on

2013-03-22 15:58:56

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

10

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 15A57

Defines

nearly decomposable

Defines

partly decomposable

Defines

fully indecomposable

Generated on Fri Feb 9 18:28:56 2018 by

LaTeXML
Givens rotation

Givens rotation

Let

A

be an

m

×

n

matrix with

m

≥

n

and full rank (viz. rank

n

). An

orthogonal matrix

triangularization (

QR Decomposition

) consists of determining an

m

×

m

orthogonal matrix

Q

such that

Q

T

⁢

A

=

[

R

0

]

with the

n

×

n

upper triangular matrix

R

. One only has then to solve the triangular system

R

⁢

x

=

P

⁢

y

, where

P

consists of the first

n

rows of

Q

.

Householder transformations

clear whole columns except for the first element of a vector. If one wants to clear parts of a matrix one element at a time, one can use

Givens rotation

, which is particularly practical for

parallel

implementation .

A matrix

G

=

[

1

⋯

0

⋯

0

⋯

0

⋮

⋱

⋮

⋮

⋮

0

⋯

c

⋯

s

⋯

0

⋮

⋮

⋱

⋮

⋮

0

⋯

-

s

⋯

c

⋯

0

⋮

⋮

⋮

⋱

⋮

0

⋯

0

⋯

0

⋯

1

]

with properly chosen

c

=

cos

⁡

(

φ

)

and

s

=

sin

⁡

(

φ

)

for some

rotation

angle

φ

can be used to zero the element

a

k

⁢

i

. The elements can be zeroed column by column from the bottom up in the following order:

(

m

,

1

)

,

(

m

,

-

1

,

1

)

,

…

,

(

2

,

1

)

,

(

m

,

2

)

,

…

,

(

3

,

2

)

,

…

,

(

m

,

n

)

,

…

,

(

n

+

1

,

n

)

.

Q

is then the product of

g

=

(

2

⁢

m

-

n

-

1

)

⁢

n

2

Givens matrices

Q

=

G

1

⁢

G

2

⁢

⋯

⁢

G

g

.

To annihilate the bottom element of a

2

×

1

vector:

(

c

s

-

s

c

)

T

⁢

(

a

b

)

=

(

r

0

)

the conditions

s

⁢

a

+

c

⁢

b

=

0

and

c

2

+

s

2

=

1

give:

c

=

a

a

2

+

b

2

,

s

=

b

a

2

+

b

2

For “Fast Givens”, see [Golub89].

References

•

Originally from The Data Analysis Briefbook
(

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

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

Golub89

Gene H. Golub and Charles F. van Loan: Matrix

Computations

, 2nd edn., The John Hopkins University Press, 1989.

Title

Givens rotation

Canonical name

GivensRotation

Date of creation

2013-03-22 12:06:10

Last modified on

2013-03-22 12:06:10

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

8

Author

akrowne (2)

Entry type

Algorithm

Classification

msc 15A57

Classification

msc 65F25

Related topic

GramSchmidtOrthogonalization

Generated on Fri Feb 9 18:28:59 2018 by

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Hermitian matrix

Hermitian matrix

For a

complex matrix

A

, let

A

∗

=

A

¯

T

, where

A

T

is the

transpose

, and

A

¯

is the

complex conjugate

of

A

.

Definition

A

complex

square matrix

A

is

Hermitian

, if

A

=

A

*

.

Properties

1.

The

eigenvalues

of a

Hermitian matrix

are real.

2.

The

diagonal

elements of a Hermitian matrix are real.

3.

The complex conjugate of a Hermitian matrix is a Hermitian matrix.

4.

If

A

is a Hermitian matrix, and

B

is a complex matrix
of same order as

A

, then

B

⁢

A

⁢

B

∗

is a Hermitian matrix.

5.

A matrix is

symmetric

if and only if it is real and Hermitian.

6.

Hermitian matrices are a

vector subspace

of the

vector space

of
complex matrices.
The real

symmetric matrices

are a

subspace

of the Hermitian matrices.

7.

Hermitian matrices are also called

self-adjoint

since if

A

is
Hermitian, then in the usual

inner product

of

ℂ

n

, we have

⟨

u

,

A

⁢

v

⟩

=

⟨

A

⁢

u

,

v

⟩

for all

u

,

v

∈

ℂ

n

.

Example

1.

For any

n

×

m

matrix

A

, the

n

×

n

matrix

A

⁢

A

∗

is
Hermitian.

2.

For any square matrix

A

, the

Hermitian part

of

A

,

1

2

⁢

(

A

+

A

∗

)

is Hermitian.
See this page (

http://planetmath.org/DirectSumOfHermitianAndSkewHermitianMatrices

).

3.

[

1

1

+

i

1

+

2

⁢

i

1

+

3

⁢

i

1

-

i

2

2

+

2

⁢

i

2

+

3

⁢

i

1

-

2

⁢

i

2

-

2

⁢

i

3

3

+

3

⁢

i

1

-

3

⁢

i

2

-

3

⁢

i

3

-

3

⁢

i

4

]

The first two examples are also examples of

normal matrices

.

Notes

1.

Hermitian matrices are named after Charles Hermite (1822-1901)

[

2

]

, who proved in 1855 that the
eigenvalues of these matrices are always real

[

1

]

.

2.

Hermitian, or self-adjoint operators on a

Hilbert space

play a fundamental
role in quantum theories as their eigenvalues are observable, or measurable; such
Hermitian operators can be represented by Hermitian matrices.

References

1

H. Eves,

Elementary Matrix

Theory

,
Dover publications, 1980.

2

The MacTutor History of Mathematics archive,

http://www-gap.dcs.st-and.ac.uk/ history/Mathematicians/Hermite.html

Charles Hermite

Title

Hermitian matrix

Canonical name

HermitianMatrix

Date of creation

2013-03-22 12:12:00

Last modified on

2013-03-22 12:12:00

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

21

Author

matte (1858)

Entry type

Definition

Classification

msc 15A57

Synonym

Hermitian

Synonym

self-adjoint

Related topic

SelfDual

Related topic

SkewHermitianMatrix

Related topic

SelfAdjointOperator

Related topic

PauliMatrices

Defines

Hermitian operator

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Hilbert matrix

Hilbert matrix

1

Hilbert Matrix

A

Hilbert matrix

H

of order

n

is a

square matrix

defined by

H

i

⁢

j

=

1

i

+

j

-

1

An example of a Hilbert matrix when

n

=

5

is

[

1

1

1

2

1

3

1

4

1

5

1

2

1

3

1

4

1

5

1

6

1

3

1

4

1

5

1

6

1

7

1

4

1

5

1

6

1

7

1

8

1

5

1

6

1

7

1

8

1

9

]

Hilbert matrices are

ill-conditioned

.

2

Inverse

The

inverse

of a Hilbert matrix

H

-

1

∈

M

N

⁢

(

ℝ

)

is given by

H

i

⁢

j

-

1

=

(

-

1

)

i

+

j

⁢

(

i

+

j

-

1

)

⁢

(

N

+

i

-

1

N

-

j

)

⁢

(

N

+

j

-

1

N

-

i

)

⁢

(

i

+

j

-

2

i

-

1

)

2

An example of an inverted Hilbert matrix when

n

=

5

case is:

[

25

-

300

1050

-

1400

630

-

300

4800

-

18900

26880

-

12600

1050

-

18900

79380

-

117600

56700

-

1400

26880

-

117600

179200

-

88200

630

-

12600

56700

-

88200

44100

]

For more fun with Hilbert matrices, see

[

1

]

.

References

1

Choi, Man-Duen. Tricks or Treats with the Hilbert Matrix. American Mathematical Monthly 90, 301-312, 1983.

Title

Hilbert matrix

Canonical name

HilbertMatrix

Date of creation

2013-03-22 13:04:14

Last modified on

2013-03-22 13:04:14

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

6

Author

Daume (40)

Entry type

Definition

Classification

msc 15A57

Classification

msc 15A09

Classification

msc 15A12

Classification

msc 65F35

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