aarjaneiro/planetmath huggingface dataset:
compound matrix

compound matrix

Suppose that

A

is an

m

×

n

matrix with entries from a field

F

and

1

≤

r

≤

min

⁡

(

m

,

n

)

. The

r

t

⁢

h

compound matrix

or

r

t

⁢

h

of

A

is the

(

m

r

)

×

(

n

r

)

matrix whose entries are

det

A

[

α

,

β

]

)

,

α

∈

Q

r

,

m

and

β

∈

Q

r

,

n

, arranged in

lexicographic order

and we use

submatrix notation

.
The notation for this matrix is

C

r

⁢

(

A

)

.

Properties

1.

C

r

⁢

(

A

⁢

B

)

=

C

r

⁢

(

A

)

⁢

C

r

⁢

(

B

)

when

r

is less than or equal to the number of rows or columns of

A

and

B

2.

If

A

is

nonsingular

, the

C

r

⁢

(

A

)

-

1

=

C

r

⁢

(

A

-

1

)

.

3.

If

A

has complex entries, then

C

r

⁢

(

A

*

)

=

(

C

r

⁢

(

A

)

)

*

.

4.

C

r

⁢

(

A

T

)

=

(

C

r

⁢

(

A

)

)

T

5.

C

r

⁢

(

A

¯

)

=

C

r

⁢

(

A

)

¯

6.

For any

k

∈

F

C

r

⁢

(

k

⁢

A

)

=

k

r

⁢

C

r

⁢

(

A

)

7.

C

r

⁢

(

I

n

)

=

I

(

n

r

)

8.

det

⁡

(

C

r

⁢

(

A

)

)

=

det

⁡

(

A

)

(

n

-

1

r

-

1

)

(Sylvester — Franke

theorem

)

Title

compound matrix

Canonical

name

CompoundMatrix

Date of creation

2013-03-22 16:13:39

Last modified on

2013-03-22 16:13:39

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

9

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 15-00

Defines

rth adjugate

Defines

Sylvester -Franke theorem

Generated on Fri Feb 9 18:13:42 2018 by

LaTeXML
conformal partitioning

conformal partitioning

Let

R

be a ring.
Let the matrices

A

∈

M

m

,

n

⁢

(

R

)

and

B

∈

M

n

,

p

⁢

(

R

)

be partitioned into

submatrices

A

i

,

j

and

B

i

,

j

respectively as follows:

A

=

n

1

n

2

⋯

n

h

[

A

1

,

1

⏞

A

1

,

2

⏞

⋯

A

1

,

h

⏞

A

2

,

1

A

2

,

2

⋯

A

2

,

h

⋮

⋮

⋮

A

g

,

1

A

g

,

2

⋯

A

g

,

h

]

}

m

1

}

m

2

⋮

}

m

g

where

A

i

,

j

is

m

i

×

n

j

,

∑

i

=

1

g

m

i

=

m

,

∑

j

=

1

h

n

j

=

n

;

B

=

p

1

p

2

⋯

p

k

[

B

1

,

1

⏞

B

1

,

2

⏞

⋯

B

1

,

k

⏞

B

2

,

1

B

2

,

2

⋯

B

2

,

k

⋮

⋮

⋮

B

h

,

1

B

h

,

2

⋯

B

h

,

k

]

}

n

1

}

n

2

⋮

}

n

h

where

B

i

,

j

is

n

i

×

p

j

,

∑

j

=

1

k

p

j

=

p

. Then

A

and

B

(in this ) are said to be

conformally partitioned

for

multiplication

.

Now suppose that

A

and

B

are conformally partitioned for multiplication.
Let

C

=

A

⁢

B

be partitioned as follows:

C

=

p

1

p

2

⋯

p

k

[

C

1

,

1

⏞

C

1

,

2

⏞

⋯

C

1

,

k

⏞

C

2

,

1

C

2

,

2

⋯

C

2

,

k

⋮

⋮

⋮

C

g

,

1

C

g

,

2

⋯

C

g

,

k

]

}

m

1

}

m

2

⋮

}

m

g

where

C

i

,

j

is

m

i

×

p

j

,

i

=

1

,

⋯

,

g

,

j

=

1

,

⋯

,

k

.
Then

C

i

,

j

=

∑

t

=

1

k

A

i

,

t

⁢

B

t

,

j

,

i

=

1

,

⋯

,

g

,

j

=

1

,

⋯

,

k

.

This method of computing

A

⁢

B

is sometimes called

block multiplication

.

Title

conformal partitioning

Canonical

name

ConformalPartitioning

Date of creation

2013-03-22 16:04:16

Last modified on

2013-03-22 16:04:16

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

9

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 15-00

Defines

block multiplication

Generated on Fri Feb 9 18:13:45 2018 by

LaTeXML
conjugate transpose

conjugate transpose

Definition

If

A

is a

complex matrix

, then the

conjugate transpose

A

∗

is the matrix

A

∗

=

A

¯

T

, where

A

¯

is
the

complex conjugate

of

A

, and

A

T

is the

transpose

of

A

.

It is clear that for

real matrices

, the conjugate transpose coincides with
the transpose.

0.0.1

Properties

1.

If

A

and

B

are complex matrices of same size, and

α

,

β

are

complex

constants, then

(

α

⁢

A

+

β

⁢

B

)

∗

=

α

¯

⁢

A

∗

+

β

¯

⁢

B

∗

,

A

∗

∗

=

A

.

2.

If

A

and

B

are complex matrices such that

A

⁢

B

is defined, then

(

A

⁢

B

)

∗

=

B

∗

⁢

A

∗

.

3.

If

A

is a complex

square matrix

, then

det

⁡

(

A

∗

)

=

det

⁡

A

¯

,

trace

⁡

(

A

∗

)

=

trace

⁡

A

¯

,

(

A

∗

)

-

1

=

(

A

-

1

)

∗

,

where

trace

and

det

are the trace
and the

determinant

operators, and

-

1

is the

inverse

operator.

4.

Suppose

⟨

⋅

,

⋅

⟩

is the standard

inner product

on

ℂ

n

.
Then for an arbitrary complex

n

×

n

matrix

A

,
and vectors

x

,

y

∈

ℂ

n

, we have

⟨

A

⁢

x

,

y

⟩

=

⟨

x

,

A

∗

⁢

y

⟩

.

Notes

The conjugate transpose of

A

is also called the

adjoint matrix

of

A

,
the

Hermitian conjugate

of

A

(whence one usually writes

A

∗

=

A

H

).
The notation

A

†

is also used for the conjugate transpose

[

2

]

.
In

[

1

]

,

A

∗

is also called the

tranjugate

of

A

.

References

1

H. Eves,

Elementary Matrix

Theory

, Dover publications, 1980.

2

M. C. Pease,

Methods of Matrix Algebra

, Academic Press, 1965.

See also

•

Wikipedia,

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

conjugate transpose

Title

conjugate transpose

Canonical name

ConjugateTranspose

Date of creation

2013-03-22 13:42:18

Last modified on

2013-03-22 13:42:18

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

10

Author

Koro (127)

Entry type

Definition

Classification

msc 15-00

Classification

msc 15A15

Synonym

adjoint matrix

Synonym

Hermitian conjugate

Synonym

tranjugate

Related topic

Transpose

Generated on Fri Feb 9 18:13:48 2018 by

LaTeXML
corollary of Schur decomposition

corollary of Schur decomposition

theorem:

A

∈

C

n

×

n

is a

normal matrix

if and only if there exists a

unitary matrix

Q

∈

ℂ

n

×

n

such that

Q

H

⁢

A

⁢

Q

=

diag

⁡

(

λ

1

,

…

,

λ

n

)

(the

diagonal matrix

)

where

H

is the

conjugate transpose

.

[

GVL

]

proof:

Firstly we show that if there exists a unitary matrix

Q

∈

ℂ

n

×

n

such that

Q

H

⁢

A

⁢

Q

=

diag

⁡

(

λ

1

,

…

,

λ

n

)

then

A

∈

ℂ

n

×

n

is a normal matrix. Let

D

=

diag

⁡

(

λ

1

,

…

,

λ

n

)

then

A

may be written as

A

=

Q

⁢

D

⁢

Q

H

. Verifying that A is normal follows by the following observation

A

⁢

A

H

=

Q

⁢

D

⁢

Q

H

⁢

Q

⁢

D

H

⁢

Q

H

=

Q

⁢

D

⁢

D

H

⁢

Q

H

and

A

H

⁢

A

=

Q

⁢

D

H

⁢

Q

H

⁢

Q

⁢

D

⁢

Q

H

=

Q

⁢

D

H

⁢

D

⁢

Q

H

. Therefore

A

is normal matrix because

D

⁢

D

H

=

diag

⁡

(

λ

1

⁢

λ

1

¯

,

…

,

λ

n

⁢

λ

n

¯

)

=

D

H

⁢

D

.

Secondly we show that if

A

∈

ℂ

n

×

n

is a normal matrix then there exists a unitary matrix

Q

∈

ℂ

n

×

n

such that

Q

H

⁢

A

⁢

Q

=

diag

⁡

(

λ

1

,

…

,

λ

n

)

. By Schur decompostion we know that there exists a

Q

∈

ℂ

n

×

n

such that

Q

H

⁢

A

⁢

Q

=

T

(

T

is an

upper triangular matrix

)

. Since

A

is a normal matrix then

T

is also a normal matrix. The result that

T

is a diagonal matrix comes from showing that a normal upper triangular matrix is

diagonal

(see

theorem for normal triangular matrices

).

QED

References

GVL

Golub, H. Gene, Van Loan F. Charles: Matrix Computations

(Third Edition)

. The Johns Hopkins University Press, London, 1996.

Title

corollary of Schur decomposition

Canonical name

CorollaryOfSchurDecomposition

Date of creation

2013-03-22 13:43:38

Last modified on

2013-03-22 13:43:38

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

7

Author

Daume (40)

Entry type

Corollary

Classification

msc 15-00

Generated on Fri Feb 9 18:13:50 2018 by

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covector

covector

If

V

is a

vector space

over a field

k

, then a

covector

is a

linear map

α

:

V

→

k

, that is, and element of the

dual space

to

V

. Thus, for example, a covector field on a

differentiable manifold

is a synonym for a 1-form (

http://planetmath.org/DifferentialForms

).

Title

covector

Canonical

name

Covector

Date of creation

2013-03-22 13:14:39

Last modified on

2013-03-22 13:14:39

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

6

Author

bwebste (988)

Entry type

Definition

Classification

msc 15-00

Generated on Fri Feb 9 18:13:51 2018 by

LaTeXML
derivation of rotation matrix using polar coordinates

derivation of rotation matrix using polar coordinates

We derive formally the expression for the

rotation

of a two-dimensional vector

𝒗

=

a

⁢

𝒙

+

b

⁢

𝒚

by an angle

ϕ

counter-clockwise. Here

𝒙

and

𝒚

are

perpendicular

unit vectors

that are oriented counter-clockwise
(the usual

orientation

).

In terms of

polar coordinates

,

𝒗

may be rewritten:

𝒗

=

r

⁢

(

cos

⁡

θ

⁢

𝒙

+

sin

⁡

θ

⁢

𝒚

)

,

a

=

r

⁢

cos

⁡

θ

;

b

=

r

⁢

sin

⁡

θ

,

for some angle

θ

and radius

r

≥

0

.
To rotate a vector

𝒗

by

ϕ

really means to shift its

polar angle

by a constant amount

ϕ

but leave its polar radius fixed.
Therefore, the result of the rotation must be:

𝒗

′

=

r

⁢

(

cos

⁡

(

θ

+

ϕ

)

⁢

𝒙

+

sin

⁡

(

θ

+

ϕ

)

⁢

𝒚

)

Expanding using the angle

addition formulae

, we obtain

𝒗

′

=

r

(

cos

θ

cos

ϕ

-

sin

θ

sin

ϕ

)

𝒙

+

(

sin

θ

cos

ϕ

+

cos

θ

sin

ϕ

)

𝒚

)

=

(

a

⁢

cos

⁡

ϕ

-

b

⁢

sin

⁡

ϕ

)

⁢

𝒙

+

(

b

⁢

cos

⁡

ϕ

+

a

⁢

sin

⁡

ϕ

)

⁢

𝒚

.

When this

transformation

is written out in

[

𝒙

,

𝒚

]

-

coordinates

, we obtain the formula for the

rotation matrix

:

𝒗

′

=

[

cos

⁡

ϕ

-

sin

⁡

ϕ

sin

⁡

ϕ

cos

⁡

ϕ

]

⁢

[

a

b

]

.

Title

derivation of rotation matrix using polar coordinates

Canonical name

DerivationOfRotationMatrixUsingPolarCoordinates

Date of creation

2013-03-22 15:25:02

Last modified on

2013-03-22 15:25:02

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

9

Author

stevecheng (10074)

Entry type

Derivation

Classification

msc 15-00

Related topic

RotationMatrix

Related topic

PolarCoordinates

Related topic

DecompositionOfOrthogonalOperatorsAsRotationsAndReflections

Related topic

DerivationOf2DReflectionMatrix

Generated on Fri Feb 9 18:13:58 2018 by

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derivative of homogeneous function

derivative of homogeneous function

Theorem 1

.

Suppose

f

:

R

n

→

R

m

is a

differentiable

positively homogeneous function of degree

r

.
Then

∂

⁡

f

∂

⁡

x

i

is a
positively homogeneous function of degree

r

-

1

.

Proof.

By considering

component

functions

if necessary, we can
assume that

m

=

1

.
For

λ

∈

ℝ

, let

M

λ

be the

multiplication

map,

M

λ

:

ℝ

n

→

ℝ

n

v

↦

λ

⁢

v

.

For

λ

>

0

and

v

∈

ℝ

n

, we have

∂

⁡

f

∂

⁡

x

i

⁢

(

λ

⁢

v

)

=

∂

⁡

(

f

∘

M

λ

∘

M

1

/

λ

)

∂

⁡

x

i

⁢

(

λ

⁢

v

)

=

∑

l

=

1

n

∂

⁡

(

f

∘

M

λ

)

∂

⁡

x

l

⁢

(

v

)

⁢

∂

(

x

↦

x

/

λ

)

l

∂

⁡

x

i

⁢

(

λ

⁢

v

)

=

∂

⁡

(

f

∘

M

λ

)

∂

⁡

x

i

⁢

(

v

)

⁢

1

λ

=

λ

r

-

1

⁢

∂

⁡

f

∂

⁡

x

i

⁢

(

v

)

as claimed.
∎

Title

derivative of homogeneous function

Canonical

name

DerivativeOfHomogeneousFunction

Date of creation

2013-03-22 14:45:05

Last modified on

2013-03-22 14:45:05

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

9

Author

matte (1858)

Entry type

Theorem

Classification

msc 15-00

Generated on Fri Feb 9 18:14:01 2018 by

LaTeXML
determinant of anti-diagonal matrix

determinant of anti-diagonal matrix

Let

A

=

adiag

⁡

(

a

1

,

…

,

a

n

)

be an

anti-diagonal matrix

. Using the sum over all

permutations

formula for the

determinant

of a matrix and since all but possibly the anti-diagonal elements are null we get directly at the result

det

⁡

A

=

sgn

⁡

(

n

,

n

-

1

,

…

,

1

)

⁢

∏

i

=

1

n

a

i

so all that remains is to calculate the sign of the permutation.
This can be done directly.

To bring the last element to the beginning

n

-

1

permutations are needed so

sgn

⁡

(

n

,

n

-

1

,

…

,

1

)

=

(

-

1

)

n

-

1

⁢

sgn

⁡

(

1

,

n

,

n

-

1

,

⋯

,

2

)

Now bring the last element to the second position.
To do this

n

-

2

permutations are needed.
Repeat this procedure

n

-

1

times to get the permutation

(

1

,

…

,

n

)

which has positive sign.

Summing every permutation, it takes

∑

k

=

1

n

-

1

k

=

n

⁢

(

n

-

1

)

2

permutations to get to the desired permutation.

So we get the final result that

det

⁡

adiag

⁡

(

a

1

,

…

,

a

n

)

=

(

-

1

)

n

⁢

(

n

-

1

)

2

⁢

∏

i

=

1

n

a

i

Notice that the sign is positive if either

n

or

n

-

1

is a multiple of

4

and negative otherwise.

Title

determinant of anti-diagonal matrix

Canonical name

DeterminantOfAntidiagonalMatrix

Date of creation

2013-03-22 15:50:25

Last modified on

2013-03-22 15:50:25

Owner

cvalente (11260)

Last modified by

cvalente (11260)

Numerical id

6

Author

cvalente (11260)

Entry type

Result

Classification

msc 15-00

Generated on Fri Feb 9 18:14:03 2018 by

LaTeXML
diagonal matrix

diagonal matrix

Definition

Let

A

be a

square matrix

(with entries in any field).
If all

off-diagonal entries

of

A

are zero, then

A

is a

diagonal matrix

.

From the definition, we see that an

n

×

n

diagonal matrix is
completely determined by the

n

entries on the

diagonal

; all other entries
are zero. If the diagonal entries are

a

1

,

a

2

,

…

,

a

n

,
then we denote the corresponding diagonal matrix by

diag

⁡

(

a

1

,

…

,

a

n

)

=

(

a

1

0

0

⋯

0

0

a

2

0

⋯

0

0

0

a

3

⋯

0

⋮

⋮

⋮

⋱

0

0

0

a

n

)

.

Examples

1.

The

identity matrix

and

zero matrix

are diagonal matrices. Also,
any

1

×

1

matrix is a diagonal matrix.

2.

A matrix

A

is a diagonal matrix if and only if

A

is
both an upper and

lower triangular matrix

.

Properties

1.

If

A

and

B

are diagonal matrices of same order, then

A

+

B

and

A

⁢

B

are again a diagonal matrix. Further, diagonal matrices
commute, i.e.,

A

⁢

B

=

B

⁢

A

. It follows that real (and

complex

)
diagonal matrices are

normal matrices

.

2.

A square matrix is diagonal if and only if it is
triangular and normal (see this page (

http://planetmath.org/TheoremForNormalTriangularMatrices

)).

3.

The

eigenvalues

of a diagonal matrix

A

=

diag

⁡

(

a

1

,

…

,

a

n

)

are

a

1

,

…

,

a

n

.
Corresponding

eigenvectors

are the standard unit vectors in

ℝ

n

.
For the

determinant

, we have

det

⁡

A

=

a

1

⁢

a

2

⁢

⋯

⁢

a

n

, so

A

is

invertible

if and only if all

a

i

are non-zero.
Then the

inverse

is given by

(

diag

⁡

(

a

1

,

…

,

a

n

)

)

-

1

=

diag

⁡

(

1

/

a

1

,

…

,

1

/

a

n

)

.

4.

If

A

is a diagonal matrix, then the adjugate of

A

is also a diagonal matrix.

5.

The

matrix exponential

of a diagonal matrix is

e

diag

⁡

(

a

1

,

…

,

a

n

)

=

diag

⁡

(

e

a

1

,

…

,

e

a

n

)

.

More generally, every

analytic function

of a diagonal matrix can be computed entrywise, i.e.:

f

⁢

(

diag

⁡

(

a

11

,

a

22

,

…

,

a

n

⁢

n

)

)

=

diag

⁡

(

f

⁢

(

a

11

)

,

f

⁢

(

a

22

)

,

…

,

f

⁢

(

a

n

⁢

n

)

)

Remarks

Diagonal matrices are also sometimes called

quasi-scalar matrices

[

1

]

.

References

1

H. Eves,

Elementary Matrix

Theory

,
Dover publications, 1980.

2

Wikipedia,

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

diagonal matrix.

Title

diagonal matrix

Canonical

name

DiagonalMatrix

Date of creation

2013-03-22 13:43:32

Last modified on

2013-03-22 13:43:32

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

12

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 15-00

Classification

msc 15A57

Synonym

quasi-scalar matrix

Synonym

quasi-scalar matrices

Synonym

diagonal matrices

Related topic

DiagonalizationLinearAlgebra

Generated on Fri Feb 9 18:14:05 2018 by

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diagonalization

diagonalization

Let

V

be a

finite-dimensional

linear space

over a field

K

, and

T

:

V

→

V

a

linear transformation

. To

diagonalize

T

is to find a basis of

V

that consists of

eigenvectors

. The

transformation

is called

diagonalizable

if such a basis exists.
The choice of terminology

reflects

the fact that
the matrix of a linear transformation relative to a given basis is

diagonal

if and only if that basis consists of
eigenvectors.

Next, we give necessary and sufficient conditions for

T

to be
diagonalizable. For

λ

∈

K

set

E

λ

=

{

u

∈

V

:

T

⁢

u

=

λ

⁢

u

}

.

It isn’t hard to show that

E

λ

is a

subspace

of

V

, and that this subspace is
non-trivial if and only if

λ

is an

eigenvalue

of

T

. In that case,

E

λ

is called the

eigenspace

associated to

λ

.

Proposition 1

A transformation is diagonalizable if and only if

dim

⁡

V

=

∑

λ

dim

⁡

E

λ

,

where the sum is taken over all eigenvalues of the transformation.

The Matrix Approach.

As was already mentioned, the term “diagonalize” comes from a matrix-based perspective. Let

M

be a

matrix representation

(

http://planetmath.org/matrix

) of

T

relative to some basis

B

. Let

P

=

[

v

1

,

…

,

v

n

]

,

n

=

dim

⁡

V

,

be a matrix whose

column vectors

are eigenvectors expressed relative
to

B

. Thus,

M

⁢

v

i

=

λ

i

⁢

v

i

,

i

=

1

,

…

,

n

where

λ

i

is the eigenvalue associated to

v

i

. The above

n

equations

are more succinctly as the

matrix equation

M

⁢

P

=

P

⁢

D

,

where

D

is the

diagonal matrix

with

λ

i

in the

i

-th
position. Now the eigenvectors in question form a basis, if and only
if

P

is

invertible

. In that case, we may write

M

=

P

⁢

D

⁢

P

-

1

.

(1)

Thus in the matrix-based approach, to “diagonalize” a matrix

M

is
to find an

invertible matrix

P

and a diagonal matrix

D

such that
equation (

1

) is satisfied.

Subtleties.

There are two fundamental reasons why a transformation

T

can fail to
be diagonalizable.

1.

The

characteristic polynomial

of

T

does not factor into linear
factors over

K

.

2.

There exists an eigenvalue

λ

, such that the kernel of

(

T

-

λ

⁢

I

)

2

is strictly greater than the kernel of

(

T

-

λ

⁢

I

)

. Equivalently, there exists an

invariant subspace

where

T

acts as a

nilpotent transformation

plus some

multiple

of the

identity

. Such subspaces manifest as non-trivial

Jordan blocks

in the

Jordan canonical form

of the transformation.

Title

diagonalization

Canonical name

Diagonalization

Date of creation

2013-03-22 12:19:49

Last modified on

2013-03-22 12:19:49

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

16

Author

rmilson (146)

Entry type

Definition

Classification

msc 15-00

Related topic

Eigenvector

Related topic

DiagonalMatrix

Defines

diagonalise

Defines

diagonalize

Defines

diagonalisation

Defines

diagonalization

Generated on Fri Feb 9 18:14:07 2018 by

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diagonally dominant matrix

diagonally dominant matrix

Let

A

be a

square matrix

of order

n

with entries

a

i

⁢

j

which are real or complex.
Then

A

is said to be

diagonally dominant

if

|

a

i

⁢

i

|

≥

∑

j

=

1

,

j

≠

i

n

|

a

i

⁢

j

|

for

i

from

1

to

n

.

In addition

A

is said to be

strictly diagonally dominant

if

|

a

i

⁢

i

|

>

∑

j

=

1

,

j

≠

i

n

|

a

i

⁢

j

|

for

i

from

1

to

n

.

Title

diagonally dominant matrix

Canonical name

DiagonallyDominantMatrix

Date of creation

2013-03-22 13:47:46

Last modified on

2013-03-22 13:47:46

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

7

Author

Daume (40)

Entry type

Definition

Classification

msc 15-00

Synonym

diagonally dominant

Synonym

strictly diagonally dominant

Defines

strictly diagonally dominant matrix

Generated on Fri Feb 9 18:14:09 2018 by

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equitable matrices of order 2

equitable matrices of order

2

The most general

2

×

2

equitable matrix

is of the form

M

=

(

1

λ

1

/

λ

1

)

for some

λ

>

0

.

Let us consider the matrix

M

=

(

m

11

m

12

m

21

m

22

)

.

A necessary and sufficient condition for

M

to be an equitable matrix
is that

m

11

,

m

12

,

m

21

,

m

22

>

0

and

m

11

=

m

11

⁢

m

11

,

m

11

=

m

12

⁢

m

21

,

m

12

=

m

11

⁢

m

12

,

m

12

=

m

12

⁢

m

22

,

m

21

=

m

21

⁢

m

11

,

m

21

=

m

22

⁢

m

21

,

m

22

=

m

21

⁢

m

12

,

m

22

=

m

22

⁢

m

22

.

It follows that

m

11

=

m

22

=

1

, and

m

12

=

1

/

m

21

.

Title

equitable matrices of order

2

Canonical name

EquitableMatricesOfOrder2

Date of creation

2013-03-22 14:58:30

Last modified on

2013-03-22 14:58:30

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

4

Author

matte (1858)

Entry type

Example

Classification

msc 15-00

Generated on Fri Feb 9 18:14:11 2018 by

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equitable matrix

equitable matrix

Equitable matrices

have been used in economics and group theory

[

1

]

.

Definition 1

.

An

n

×

n

matrix

M

=

(

m

i

⁢

j

)

is an

equitable matrix

if all

m

i

⁢

j

are

positive

, and

m

i

⁢

j

=

m

i

⁢

k

⁢

m

k

⁢

j

for all

i

,

j

,

k

=

1

,

…

,

n

.

Setting

i

=

j

=

k

yields

m

i

⁢

i

=

m

i

⁢

i

⁢

m

i

⁢

i

so

diagonal

elements of
equitable matrices equal

1

. Next, setting

i

=

j

yields

m

i

⁢

i

=

m

i

⁢

k

⁢

m

k

⁢

i

, so

m

i

⁢

k

=

1

/

m

k

⁢

i

.

Examples

1.

An example of an equitable matrix of order

n

is

(

1

⋯

1

⋮

⋱

⋮

1

⋯

1

)

.

This example shows that equitable matrices exist for all

n

.

2.

The most general equitable matrix of orders

2

and

3

are

(

1

a

1

/

a

1

)

,

and

(

1

a

a

⁢

b

1

/

a

1

b

1

/

a

⁢

b

1

/

b

1

)

,

where

a

,

b

,

c

>

0

.

Properties

1.

A

n

×

n

matrix

M

=

(

m

i

⁢

j

)

is equitable if and
only if it can be expressed in the form

m

i

⁢

j

=

exp

⁡

(

λ

i

-

λ

j

)

for real numbers

λ

1

,

λ

2

,

…

,

λ

n

with

λ

1

=

0

. (proof. (

http://planetmath.org/ParameterizationOfEquitableMatrices

))

2.

An equitable matrix is completely determined by its first row.
If

m

1

⁢

i

,

i

=

1

,

…

,

n

are known, then

m

i

⁢

j

=

m

1

⁢

j

m

1

⁢

i

.

3.

If

M

is an

n

×

n

equitable matrix, then

exp

⁡

(

M

)

=

I

+

e

n

-

1

n

⁢

M

,

where

exp

is the

matrix exponential

.

4.

Equitable matrices form a group under the

Hadamard product

[

1

]

.

5.

If

M

is an

n

×

n

equitable matrix and

s

:

{

1

,

…

,

r

}

→

{

1

,

…

,

n

}

is a mapping, then

K

a

⁢

b

=

M

s

⁢

(

a

)

⁢

s

⁢

(

b

)

,

a

,

b

=

1

,

…

,

r

is an equitable

r

×

r

matrix. In particular, striking the

l

:th row and column in an
equitable matrix yields a new equitable matrix.

See

[

1

]

for further

properties

and references.

References

1

H. Eves,

Elementary Matrix

Theory

, Dover publications, 1980.

Title

equitable matrix

Canonical

name

EquitableMatrix

Date of creation

2013-03-22 14:58:28

Last modified on

2013-03-22 14:58:28

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

11

Author

matte (1858)

Entry type

Definition

Classification

msc 15-00

Related topic

HadamardProduct

Generated on Fri Feb 9 18:14:13 2018 by

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exactly determined

exactly determined

An

exactly determined

system of linear equations

has precisely as many unknowns as equations and is hence soluble.

Title

exactly determined

Canonical name

ExactlyDetermined

Date of creation

2013-03-22 12:06:58

Last modified on

2013-03-22 12:06:58

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

5

Author

akrowne (2)

Entry type

Definition

Classification

msc 15-00

Synonym

exactly-determined

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