aarjaneiro/planetmath huggingface dataset:
Levy-Desplanques theorem

Levy-Desplanques theorem

A

strictly diagonally dominant matrix

is

non-singular

. In other words, let

A

∈

𝐂

n

,

n

be a matrix satisfying the

property

|

a

i

⁢

i

|

>

∑

j

≠

i

|

a

i

⁢

j

|

∀

i

;

then

det

⁡

(

A

)

≠

0

.

Proof:
Let

det

⁡

(

A

)

=

0

; then a non-zero vector

𝐱

exists such that

A

⁢

𝐱

=

𝟎

; let

M

be the index such that

|

x

M

|

=

max

⁡

(

|

x

1

|

,

|

x

2

|

,

⋯

,

|

x

n

|

)

, so that

|

x

j

|

≤

|

x

M

|

∀

j

; we have

a

M

⁢

1

⁢

x

1

+

a

M

⁢

2

⁢

x

2

+

⋯

+

a

M

⁢

M

⁢

x

M

+

⋯

+

a

M

⁢

n

⁢

x

n

=

0

which implies:

|

a

M

⁢

M

|

⁢

|

x

M

|

=

|

a

M

⁢

M

⁢

x

M

|

=

|

∑

j

≠

M

a

M

⁢

j

⁢

x

j

|

≤

∑

j

≠

M

|

a

M

⁢

j

|

⁢

|

x

j

|

≤

|

x

M

|

⁢

∑

j

≠

M

|

a

M

⁢

j

|

that is

|

a

M

⁢

M

|

≤

∑

j

≠

M

|

a

M

⁢

j

|

,

in contrast with

strictly

diagonally dominance

definition

.

□

Remark:
the

Levy-Desplanques theorem

is

equivalent

to the well-known

Gerschgorin circle theorem

. In fact, let’s assume Levy-Desplanques theorem is true, and let

A

a

n

×

n

complex-valued matrix, with an

eigenvalue

λ

; let’s apply Levy-Desplanques theorem to the matrix

B

=

A

-

λ

⁢

I

, which is

singular

by definition of eigenvalue: an index

i

must exist for which

|

a

i

⁢

i

-

λ

|

=

|

b

i

⁢

i

|

≤

∑

j

≠

i

n

|

b

i

⁢

j

|

=

∑

j

≠

i

n

|

a

i

⁢

j

|

, which is Gerschgorin circle theorem.
On the other hand, let’s assume Gerschgorin circle theorem is true, and let

A

be a strictly diagonally dominant

n

×

n

complex matrix

. Then, since the

absolute value

of each disc center

|

a

i

⁢

i

|

is strictly greater than the same disc radius

∑

j

≠

i

n

|

a

i

⁢

j

|

, the point

λ

=

0

can’t belong to any circle, so it doesn’t belong to the

spectrum

of

A

, which therefore can’t be singular.

Title

Levy-Desplanques theorem

Canonical

name

LevyDesplanquesTheorem

Date of creation

2013-03-22 15:34:50

Last modified on

2013-03-22 15:34:50

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

9

Author

Andrea Ambrosio (7332)

Entry type

Theorem

Classification

msc 15-00

Generated on Fri Feb 9 18:15:04 2018 by

LaTeXML
linear complex structure

linear complex structure

A

on a

real vector space

V

, with

dim

⁢

(

V

)

=

m

, is a linear

automorphism

J

∈

Aut

⁢

(

V

)

such that

J

2

=

J

∘

J

=

-

id

V

. With a complex structure

J

we can consider

V

as a

complex vector space

with the

product

ℂ

×

V

→

V

given by

(

x

+

i

⁢

y

)

⁢

𝐯

=

x

⁢

𝐯

+

y

⁢

J

⁢

(

𝐯

)

,

∀

x

,

y

∈

ℝ

,

𝐯

∈

V

.

This implies that the

dimension

m

of

V

must be even.

A common example is

V

=

ℝ

2

⁢

n

with the

standard basis

𝐞

1

,

…

,

𝐞

n

,

𝐟

1

,

…

,

𝐟

n

, for which we can obtain a complex structure

J

0

∈

Aut

⁢

(

ℝ

2

⁢

n

)

represented by the matrix

(

𝟎

𝐈

n

-

𝐈

n

𝟎

)

.

Here

𝐈

n

∈

M

n

⁢

(

ℝ

)

is the

identity

n

×

n

matrix and

𝟎

∈

M

n

⁢

(

ℝ

)

is the zero

n

×

n

matrix.

Title

linear complex structure

Canonical name

LinearComplexStructure

Date of creation

2013-03-22 16:16:48

Last modified on

2013-03-22 16:16:48

Owner

Mazzu (14365)

Last modified by

Mazzu (14365)

Numerical id

12

Author

Mazzu (14365)

Entry type

Definition

Classification

msc 15-00

Related topic

ComplexificationOfVectorSpace

Defines

linear complex structure

Generated on Fri Feb 9 18:15:08 2018 by

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linear extension

linear extension

Let

R

be a

commutative ring

,

M

a free

R

-module,

B

a basis of

M

, and

N

a further

R

-module. Each

element

m

∈

M

then has a
unique

representation

m

=

∑

b

∈

B

m

b

⁢

b

,

where

m

b

∈

R

for all

b

∈

B

, and only finitely many

m

b

are
non-zero. Given a set map

f

1

:

B

→

N

we may therefore define the

R

-

module homomorphism

φ

1

:

M

→

N

, called the

linear extension

of

f

1

, such that

m

↦

∑

b

∈

B

m

b

⁢

f

1

⁢

(

b

)

.

The map

φ

1

is the unique

homomorphism

from

M

to

N

whose

restriction

to

B

is

f

1

.

The above observation has a convenient reformulation in terms of

category theory

. Let

𝖱𝖬𝗈𝖽

denote the

category

of

R

-modules, and

𝖲𝖾𝗍

the

category of sets

. Consider the

adjoint functors

U

:

𝖱𝖬𝗈𝖽

→

𝖲𝖾𝗍

, the

forgetful functor

that maps an

R

-module to its underlying set, and

F

:

𝖲𝖾𝗍

→

𝖱𝖬𝗈𝖽

,
the

free module

functor

that maps a set to the free

R

-module generated by that set. To say that

U

is right-adjoint to

F

is the same as saying that every set map from

B

to

U

⁢

(

N

)

, the set underlying

N

, corresponds naturally and bijectively to an

R

-module homomorphism from

M

=

F

⁢

(

B

)

to

N

.

Similarly, given a map

f

2

:

B

2

→

N

, we may define the

bilinear extension

φ

2

:

M

2

→

N

(

m

,

n

)

↦

∑

b

∈

B

∑

c

∈

B

m

b

⁢

n

c

⁢

f

2

⁢

(

b

,

c

)

,

which is the unique

bilinear map

from

M

2

to

N

whose restriction
to

B

2

is

f

2

.

Generally, for any positive integer

n

and a map

f

n

:

B

n

→

N

, we may define the

n

-linear extension

φ

n

:

M

n

→

N

m

↦

∑

b

∈

B

n

m

b

⁢

f

n

⁢

(

b

)

quite compactly using

multi-index notation

:

m

b

=

∏

k

=

1

n

m

k

,

b

k

.

Usage

The notion of linear extension is typically used as a

manner-of-speaking

. Thus, when a multilinear map is defined
explicitly in a mathematical text, the images of the basis elements
are given accompanied by the phrase “by multilinear extension” or

similar

.

Title

linear extension

Canonical name

LinearExtension

Date of creation

2013-03-22 15:24:06

Last modified on

2013-03-22 15:24:06

Owner

GrafZahl (9234)

Last modified by

GrafZahl (9234)

Numerical id

7

Author

GrafZahl (9234)

Entry type

Definition

Classification

msc 15-00

Related topic

basis

Related topic

Basis

Defines

bilinear extension

Defines

multilinear extension

Defines

n

-linear extension

Generated on Fri Feb 9 18:15:11 2018 by

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linear least squares

linear least squares

Let

A

be an

m

×

n

matrix with

m

≥

n

and

b

an

m

×

1

matrix. We want to consider the problem

A

⁢

x

≈

b

where

≈

stands for the best approximate

solution

in the least squares sense, i.e. we want to minimize the

Euclidean norm

of the residual

r

=

A

⁢

x

-

b

||

A

⁢

x

-

b

||

2

=

||

r

||

2

=

[

∑

i

=

1

m

r

i

2

]

1

/

2

We want to find the vector

x

which is closest to

b

in the

column space

of

A

.

Among the different methods to solve this problem, we mention

Normal Equations

, sometimes

ill-conditioned

,

QR Decomposition

, and, most generally,

Singular Value Decomposition

. For further reading, [Golub89], [Branham90], [Wong92], [Press95].

Example: Let us consider the problem of finding the closest point (vertex) to measurements on

straight lines

(e.g.

trajectories

emanating from a particle collision). This problem can be described by

A

⁢

x

=

b

with

A

=

[

a

11

a

12

⋮

⋮

a

m

⁢

1

a

m

⁢

2

]

;

x

=

[

u

v

]

;

b

=

[

b

1

⋮

b

m

]

This is clearly an

inconsistent

system of linear equations

, with more equations than unknowns, a frequently occurring problem in experimental data analysis. The system is, however, not very inconsistent and there is a point that lies “nearly” on all straight lines. The solution can be found with the

linear least squares

method, e.g. by QR decomposition for solving

A

⁢

x

=

b

:

Q

⁢

R

⁢

x

=

b

→

x

=

R

-

1

⁢

Q

T

⁢

b

References

•

Originally from The Data Analysis Briefbook
(

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

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

Wong92

S.S.M. Wong, Computational Methods in Physics and Engineering, Prentice Hall, 1992.

Golub89

Gene H. Golub and Charles F. van Loan: Matrix Computations, 2nd edn., The John Hopkins University Press, 1989.

Branham90

R.L. Branham, Scientific Data Analysis, An Introduction to Overdetermined Systems, Springer, Berlin, Heidelberg, 1990.

Press95

W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C, Second edition, Cambridge University Press, 1995. (The same book exists for the Fortran language). There is also an Internet version which you can work from.

Title

linear least squares

Canonical name

LinearLeastSquares

Date of creation

2013-03-22 12:05:50

Last modified on

2013-03-22 12:05:50

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

5

Author

akrowne (2)

Entry type

Definition

Classification

msc 15-00

Related topic

LeastSquares

Generated on Fri Feb 9 18:15:13 2018 by

LaTeXML
matrices of special form

matrices of special form

The aim of the present entry is to list matrices with special

properties

.

Restriction on form

•

diagonal matrices

•

anti-diagonal matrices

•

triangular matrices

•

block matrices

of either form above or of some other form

•

nilpotent matrices

•

elementary matrices

•

Hadamard matrices

•

partly decomposable

matrices

•

fully indecomposable matrices

•

nearly decomposable matrices

•

doubly stochastic matrices

•

stochastic matrices

In numerical applications,

sparse matrices

(matrices with few nonzero entries) are important; these can be of any of the forms above.

Other restrictions

•

singular matrices

•

GL

⁡

(

n

)

:

invertible matrices

•

SU

⁡

(

n

)

:

unitary matrices

with

determinant

1

•

O

⁡

(

n

)

:

orthogonal matrices

•

SO

⁡

(

n

)

:
orthogonal matrices with determinant

1

(

http://planetmath.org/RotationMatrix

)

•

normal matrices

•

positive definite matrices

•

symmetric matrices

,

antisymmetric matrices

,

Hermitian matrices

,

anti-Hermitian matrices

•

Sp

⁡

(

2

⁢

n

)

: symplectic matrices (also called

Sp

⁡

(

n

)

)

Special matrices

•

zero matrix

•

identity matrix

•

Hilbert matrix

•

Vandermonde matrix

•

Toeplitz matrix

•

Pascal matrix

•

Cauchy matrix

•

M-matrix

•

magic square

•

Latin square

Title

matrices of special form

Canonical

name

MatricesOfSpecialForm

Date of creation

2013-03-22 14:13:49

Last modified on

2013-03-22 14:13:49

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

15

Author

matte (1858)

Entry type

Definition

Classification

msc 15-00

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

LaTeXML
Neumann series

Neumann series

If

A

is a

square matrix

,

∥

A

∥

<

1

, then

I

-

A

is

nonsingular

and

(

I

-

A

)

-

1

=

I

+

A

+

A

2

+

⋯

=

∑

k

=

0

∞

A

k

. This is the

Neumann series

.

It provides approximations of

(

I

-

A

)

-

1

when

A

has entries of small magnitude.
For example, a first-order approximation is

(

I

-

A

)

-

1

≈

I

+

A

.

It is obvious that this is a

generalization

of the

geometric series

.

References

1

Carl D. Meyer,

Matrix Analysis and Applied

Linear Algebra

.

Title

Neumann series

Canonical name

NeumannSeries

Date of creation

2013-03-22 15:25:49

Last modified on

2013-03-22 15:25:49

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

9

Author

georgiosl (7242)

Entry type

Theorem

Classification

msc 15-00

Generated on Fri Feb 9 18:15:23 2018 by

LaTeXML
nilpotent matrix

nilpotent matrix

The

square matrix

A

is said to be

nilpotent

if

A

n

=

A

⁢

A

⁢

⋯

⁢

A

⏟

n times

=

𝟎

for some

positive integer

n

(here

𝟎

denotes the matrix where every entry is 0).

Theorem

(Characterization of nilpotent matrices).

A matrix is nilpotent iff its

eigenvalues

are all 0.

Proof.

Let

A

be a

nilpotent matrix

. Assume

A

n

=

𝟎

. Let

λ

be an eigenvalue of

A

.
Then

A

⁢

𝐱

=

λ

⁢

𝐱

for some nonzero vector

𝐱

.
By induction

λ

n

⁢

𝐱

=

A

n

⁢

𝐱

=

0

, so

λ

=

0

.

Conversely, suppose that all eigenvalues of

A

are zero. Then the chararacteristic

polynomial

of

A

:

det

⁡

(

λ

⁢

I

-

A

)

=

λ

n

. It now follows from the

Cayley-Hamilton theorem

that

A

n

=

𝟎

.
∎

Since the

determinant

is the product of the eigenvalues it follows that a nilpotent matrix has determinant 0. Similarly, since the trace of a square matrix is the sum of the eigenvalues, it follows that it has trace 0.

One class of nilpotent matrices are the

http://planetmath.org/node/4381

strictly

triangular matrices

(lower or upper), this follows from the fact that the eigenvalues of a triangular matrix are the

diagonal

elements, and thus are all zero in the case of

strictly

triangular matrices.

Note for

2

×

2

matrices

A

the theorem implies that

A

is nilpotent iff

A

=

𝟎

or

A

2

=

𝟎

.

Also it is worth noticing that any matrix that is

similar

to a nilpotent matrix is nilpotent.

Title

nilpotent matrix

Canonical name

NilpotentMatrix

Date of creation

2013-03-22 13:05:56

Last modified on

2013-03-22 13:05:56

Owner

jgade (861)

Last modified by

jgade (861)

Numerical id

17

Author

jgade (861)

Entry type

Definition

Classification

msc 15-00

Generated on Fri Feb 9 18:15:25 2018 by

LaTeXML
nilpotent transformation

nilpotent transformation

A

linear transformation

N

:

U

→

U

is called

nilpotent

if there exists a

k

∈

ℕ

such that

N

k

=

0

.

A nilpotent transformation naturally determines a flag of

subspaces

{

0

}

⊂

ker

⁡

N

1

⊂

ker

⁡

N

2

⊂

…

⊂

ker

⁡

N

k

-

1

⊂

ker

⁡

N

k

=

U

and a

signature

0

=

n

0

<

n

1

<

n

2

<

…

<

n

k

-

1

<

n

k

=

dim

⁡

U

,

n

i

=

dim

⁡

ker

⁡

N

i

.

The signature is governed by the following constraint, and
characterizes

N

up to

linear isomorphism

.

Proposition 1

A

sequence

of increasing

natural numbers

is the signature of a nil-potent

transformation

if and only if

n

j

+

1

-

n

j

≤

n

j

-

n

j

-

1

for all

j

=

1

,

…

,

k

-

1

. Equivalently, there exists a basis of

U

such that the matrix of

N

relative to this basis is block

diagonal

(

N

1

0

0

…

0

0

N

2

0

…

0

0

0

N

3

…

0

⋮

⋮

⋮

⋱

⋮

0

0

0

…

N

k

)

,

with each of the blocks having the form

N

i

=

(

0

1

0

…

0

0

0

0

1

…

0

0

⋮

⋮

⋮

⋱

⋮

0

0

0

…

1

0

0

0

0

…

0

1

0

0

0

…

0

0

)

Letting

d

i

denote the number of blocks of size

i

, the
signature of

N

is given by

n

i

=

n

i

-

1

+

d

i

+

d

i

+

1

+

…

+

d

k

,

i

=

1

,

…

,

k

Title

nilpotent transformation

Canonical name

NilpotentTransformation

Date of creation

2013-03-22 12:19:52

Last modified on

2013-03-22 12:19:52

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

7

Author

rmilson (146)

Entry type

Definition

Classification

msc 15-00

Synonym

nilpotent

Related topic

LinearTransformation

Generated on Fri Feb 9 18:15:27 2018 by

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non-zero vector

non-zero vector

A

non-zero vector

in a

vector space

V

is a vector that is not equal to
the

zero vector

in

V

.

If

𝐮

is a non-zero vector in

V

and

a

is a scalar,
i.e. an element of the field over which

V

is defined, then the

equation

a

⁢

𝐮

=

𝟎

is
possible only in the case

a

=

0

.

Title

non-zero vector

Canonical name

NonzeroVector

Date of creation

2013-12-01 16:50:29

Last modified on

2013-12-01 16:50:29

Owner

matte (1858)

Last modified by

pahio (2872)

Numerical id

6

Author

matte (2872)

Entry type

Definition

Classification

msc 15-00

Generated on Fri Feb 9 18:15:28 2018 by

LaTeXML
off-diagonal entry

off-diagonal entry

Let

A

=

(

a

i

⁢

j

)

be a

square matrix

. An element

a

i

⁢

j

is an

off-diagonal entry

if

a

i

⁢

j

is not on the

diagonal

, i.e., if

i

≠

j

.

Title

off-diagonal entry

Canonical name

OffdiagonalEntry

Date of creation

2013-03-22 13:45:26

Last modified on

2013-03-22 13:45:26

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 15-00

Classification

msc 15A57

Generated on Fri Feb 9 18:15:30 2018 by

LaTeXML
orthogonal matrices

orthogonal matrices

A real square

n

×

n

matrix

Q

is

orthogonal

if

Q

T

⁢

Q

=

I

, i.e., if

Q

-

1

=

Q

T

. The rows and columns of an

orthogonal matrix

form an

orthonormal basis

.

Orthogonal matrices play a very important role in

linear algebra

.

Inner products

are preserved under an orthogonal transform:

(

Q

⁢

x

)

T

⁢

Q

⁢

y

=

x

T

⁢

Q

T

⁢

Q

⁢

y

=

x

T

⁢

y

, and also the

Euclidean norm

||

Q

⁢

x

||

2

=

||

x

||

2

. An example of where this is useful is solving the least squares problem

A

⁢

x

≈

b

by solving the equivalent problem

Q

T

⁢

A

⁢

x

≈

Q

T

⁢

b

.

Orthogonal matrices can be thought of as the real case of

unitary matrices

. A unitary matrix

U

∈

ℂ

n

×

n

has the property

U

*

⁢

U

=

I

, where

U

*

=

U

T

¯

(the

conjugate transpose

). Since

Q

T

¯

=

Q

T

for real

Q

, orthogonal matrices are

unitary

.

An orthogonal matrix

Q

has

det

⁡

(

Q

)

=

±

1

.

Important orthogonal matrices are

Givens rotations

and

Householder transformations

. They help us maintain numerical stability because they do not amplify

rounding errors

.

Orthogonal

2

×

2

matrices are

rotations

or

reflections

if they have the form:

(

cos

⁡

(

α

)

sin

⁡

(

α

)

-

sin

⁡

(

α

)

cos

⁡

(

α

)

)

⁢

or

⁢

(

cos

⁡

(

α

)

sin

⁡

(

α

)

sin

⁡

(

α

)

-

cos

⁡

(

α

)

)

respectively.

This entry is based on content from The Data Analysis Briefbook (

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

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

References

1

Friedberg, Insell, Spence.

Linear Algebra

. Prentice-Hall Inc., 1997.

Title

orthogonal matrices

Canonical name

OrthogonalMatrices

Date of creation

2013-03-22 12:05:19

Last modified on

2013-03-22 12:05:19

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

11

Author

akrowne (2)

Entry type

Definition

Classification

msc 15-00

Related topic

OrthogonalPolynomials

Related topic

RotationMatrix

Generated on Fri Feb 9 18:15:32 2018 by

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orthogonal vectors

orthogonal vectors

Two vectors,

v

1

and

v

2

, are

orthogonal

if and only if their

inner product

⟨

x

,

y

⟩

is 0. In two

dimensions

,

orthogonal vectors

are

perpendicular

(or in

n

dimensions in the plane defined by the two vectors.)

A set of vectors is orthogonal when, taken pairwise, any two vectors in the set are orthogonal.

Title

orthogonal vectors

Canonical

name

OrthogonalVectors

Date of creation

2013-03-22 12:07:33

Last modified on

2013-03-22 12:07:33

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

8

Author

akrowne (2)

Entry type

Definition

Classification

msc 15-00

Related topic

GramSchmidtOrthogonalization

Generated on Fri Feb 9 18:15:33 2018 by

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overdetermined

overdetermined

An

overdetermined

system of linear equations

has more equations than unknowns. In general, overdetermined systems have no solution. In some cases,

linear least squares

may be used to find an approximate solution.

Title

overdetermined

Canonical name

Overdetermined

Date of creation

2013-03-22 12:07:00

Last modified on

2013-03-22 12:07:00

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

9

Author

akrowne (2)

Entry type

Definition

Classification

msc 15-00

Synonym

over-determined

Synonym

over determined

Related topic

UnderDetermined

Related topic

RankNullityTheorem

Generated on Fri Feb 9 18:15:35 2018 by

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parameterization of equitable matrices

parameterization of equitable matrices

A

n

×

n

matrix 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

.

Assume that

m

i

⁢

j

are the entries of an

equitable matrix

.

Since all the elements of an equitable matrix are

positive

by definition, we can write

m

i

⁢

j

=

exp

⁡

μ

i

⁢

j

with the quantities

μ

i

⁢

j

being real numbers (which may be positive, negative or zero).

In terms of this

representation

, the defining

identity

for an equitable matrix becomes

μ

i

⁢

k

=

μ

i

⁢

j

+

μ

j

⁢

k

Since this comprises a

system of linear equations

for the quantities

μ

i

⁢

j

, we could solve it using the usual methods of matrix theory. However, for this particular system of linear equations, there is a much simpler approach.

Consider the special case of the identity when

i

=

j

=

k

:

μ

i

⁢

i

=

μ

i

⁢

i

+

μ

i

⁢

i

.

This simplifies to

μ

i

⁢

i

=

0

.

In other words, all the

diagonal

entries are zero.

Consider the case when

i

=

k

(but does not equal

j

).

μ

i

⁢

j

+

μ

j

⁢

i

=

μ

i

⁢

i

By wat we have just shown, the

right hand side

of this equation equals zero. Hence, we have

μ

i

⁢

j

=

-

μ

j

⁢

i

.

In other words, the matrix of

μ

’s is

antisymmetric

.

We may express any entry in terms of the

n

entries

μ

i

⁢

1

:

μ

i

⁢

j

=

μ

i

⁢

1

+

μ

1

⁢

j

=

μ

i

⁢

1

-

μ

j

⁢

1

We will conclude by noting that if, given any

n

numbers

λ

i

with

λ

1

=

0

, but the remaining

λ

’s arbitrary, we define

μ

i

⁢

j

=

λ

i

-

λ

j

,

then

μ

i

⁢

j

+

μ

j

⁢

k

=

λ

i

-

λ

j

+

λ

j

-

λ

k

=

λ

i

-

λ

k

=

μ

i

⁢

k

Hence, we obtain a solution of the equations

μ

i

⁢

k

=

μ

i

⁢

j

+

μ

j

⁢

k

.

Moreover, by what we what we have seen, if we set

λ

i

=

μ

i

⁢

1

, all solutions of these equations can be so described.

Q.E.D.

Title

parameterization of equitable matrices

Canonical name

ParameterizationOfEquitableMatrices

Date of creation

2013-03-22 14:58:36

Last modified on

2013-03-22 14:58:36

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 15-00

Generated on Fri Feb 9 18:15:37 2018 by

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partitioned matrix

partitioned matrix

A

partitioned matrix

, or a

block matrix

, is
a matrix

M

that has been constructed from other smaller matrices.
These smaller matrices are called

blocks

or

sub-matrices

of

M

.

For

instance

, if we

partition

the below

5

×

5

matrix
as follows

L

=

(

1

0

1

2

3

0

1

1

2

3

2

3

9

9

9

2

3

9

9

9

2

3

9

9

9

)

,

then we can define the matrices

A

=

(

1

0

0

1

)

,

B

=

(

1

2

3

1

2

3

)

,

C

=

(

2

3

2

3

2

3

)

,

D

=

(

9

9

9

9

9

9

9

9

9

)

and write

L

as

L

=

(

A

B

C

D

)

,

or

⁢

L

=

(

A

B

C

D

)

.

If

A

1

,

…

,

A

n

are

square matrices

(of possibly
different sizes), then we define the

direct sum

of
the matrices

A

1

,

…

,

A

n

as the partitioned matrix

diag

⁡

(

A

1

,

…

,

A

n

)

=

(

A

1

⋱

A

n

)

,

where the off-diagonal blocks are zero.

If

A

and

B

are matrices of the same size partitioned into blocks of the same size, the partition of the sum is the sum of the partitions.

If

A

and

B

are

m

×

n

and

n

×

k

matrices, respectively, then if the blocks of

A

and

B

are of the correct size to be multiplied, then the blocks of the

product

are the products of the blocks.

Title

partitioned matrix

Canonical name

PartitionedMatrix

Date of creation

2013-03-22 13:32:55

Last modified on

2013-03-22 13:32:55

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

11

Author

mathcam (2727)

Entry type

Definition

Classification

msc 15-00

Related topic

JordanCanonicalForm

Related topic

JordanCanonicalFormTheorem

Defines

block matrix

Defines

sub-matrix

Defines

submatrix

Generated on Fri Feb 9 18:15:39 2018 by

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