aarjaneiro/planetmath huggingface dataset:
skew Hadamard matrix

skew Hadamard matrix

A

Hadamard matrix

H

is

skew Hadamard

if

H

+

H

T

=

2

⁢

I

.

A

collection

of

skew Hadamard matrices

, including at least one example of every order

n

≤

100

and also including every

equivalence class

of order

≤

28

, is available

http://www.rangevoting.org/SkewHad.html

at this web page.
It has been conjectured that one exists for every

positive

order divisible by 4.

Reid and Brown in 1972 showed that there exists a
“doubly

regular

tournament

of order n”
if and only if there exists a skew Hadamard matrix of order n+1.

References

1

S. Georgiou, C. Koukouvinos, J. Seberry,

Hadamard matrices,

orthogonal

designs and construction algorithms

, pp. 133-205 in DESIGNS 2002: Further computational and constructive

design theory

, Kluwer 2003.

2

K.B. Reid, E. Brown,

Doubly regular tournaments are

equivalent

to skew Hadamard matrices

, J. Combinatorial Theory A 12 (1972) 332-338.

3

J. Seberry, M.Yamada,

Hadamard matrices,

sequences

, and

block designs

, pp. 431-560 in Contemporary Design Theory, a collection of surveys (J.H.Dinitz & D.R.Stinson eds.), Wiley 1992.

Title

skew Hadamard matrix

Canonical name

SkewHadamardMatrix

Date of creation

2013-03-22 16:13:02

Last modified on

2013-03-22 16:13:02

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

13

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 15-00

Generated on Fri Feb 9 18:16:06 2018 by

LaTeXML
skew-symmetric matrix

skew-symmetric matrix

Definition:

Let

A

be an

square matrix

of
order

n

with real entries

(

a

i

⁢

j

)

.
The matrix

A

is

skew-symmetric

if

a

i

⁢

j

=

-

a

j

⁢

i

for all

1

≤

i

≤

n

,

1

≤

j

≤

n

.

A

=

(

a

11

=

0

⋯

a

1

⁢

n

⋮

⋱

⋮

a

n

⁢

1

⋯

a

n

⁢

n

=

0

)

The main diagonal entries are zero because

a

i

,

i

=

-

a

i

,

i

implies

a

i

,

i

=

0

.

One can see

skew-symmetric matrices

as a
special case of

complex

skew-Hermitian matrices

. Thus,
all properties of skew-Hermitian matrices also hold
for skew-symmetric matrices.

Properties:

1.

The matrix

A

is skew-symmetric if and only if

A

t

=

-

A

, where

A

t

is the matrix transpose

2.

For the trace operator, we have that

tr

⁡

(

A

)

=

tr

⁡

(

A

t

)

.
Combining this with property (1), it follows
that

tr

⁡

(

A

)

=

0

for a skew-symmetric matrix

A

.

3.

Skew-symmetric matrices form a

vector space

: If

A

and

B

are skew-symmetric and

α

,

β

∈

ℝ

, then

α

⁢

A

+

β

⁢

B

is also skew-symmetric.

4.

Suppose

A

is a skew-symmetric matrix and

B

is a matrix of
same order as

A

. Then

B

t

⁢

A

⁢

B

is skew-symmetric.

5.

All

eigenvalues

of skew-symmetric matrices are

purely imaginary

or zero. This result is proven on the page
for skew-Hermitian matrices.

6.

According to Jacobi’s Theorem, the

determinant

of a
skew-symmetric matrix of odd order is zero.

Examples:

•

(

0

b

-

b

0

)

•

(

0

b

c

-

b

0

e

-

c

-

e

0

)

Title

skew-symmetric matrix

Canonical name

SkewsymmetricMatrix

Date of creation

2013-03-22 12:01:05

Last modified on

2013-03-22 12:01:05

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

10

Author

Daume (40)

Entry type

Definition

Classification

msc 15-00

Related topic

SelfDual

Related topic

AntiSymmetric

Related topic

SkewHermitianMatrix

Related topic

AntisymmetricMapping

Generated on Fri Feb 9 18:16:08 2018 by

LaTeXML
square matrix

square matrix

A

square matrix

has the same number of rows as columns.

Examples:

1.

(

1.00000

0.50000

0.33333

0.25000

0.50000

0.33333

0.25000

0.20000

0.33333

0.25000

0.20000

0.16667

0.25000

0.20000

0.16667

0.14286

)

2.

(

1

)

3.

(

0.94

0.37

0.71

0.32

0.58

0.90

0.16

0.74

0.83

0.27

0.50

0.03

0.07

0.49

0.55

0.15

0.59

0.43

0.03

0.76

0.04

0.64

0.61

0.17

0.29

)

4.

(

89

38

50

64

26

98

40

96

83

)

The notation

Mat

n

⁡

(

𝕂

)

is often used to signify the set of square matrices which are of order

n

(size

n

×

n

) with elements drawn from a field

𝕂

. Thus, one would use

a

∈

Mat

3

⁡

(

ℂ

)

to declare that

a

is a three-by-three matrix with elements that are

complex numbers

.

Property

: Suppose

A

and

B

are matrices such that

A

⁢

B

is a square matrix. Then the product

B

⁢

A

is defined and also a square matrix.

Title

square matrix

Canonical name

SquareMatrix

Date of creation

2013-03-22 12:11:27

Last modified on

2013-03-22 12:11:27

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

9

Author

akrowne (2)

Entry type

Definition

Classification

msc 15-00

Generated on Fri Feb 9 18:16:10 2018 by

LaTeXML
strictly upper triangular matrix

strictly upper triangular matrix

A

strictly

is an

upper triangular matrix

which has

0

on the main diagonal. Similarly a

strictly lower triangular matrix

is a

lower triangular matrix

which has

0

on the main diagonal. i.e.

A

strictly upper triangular matrix

is of the form

[

0

a

12

a

13

⋯

a

1

⁢

n

0

0

a

23

⋯

a

2

⁢

n

0

0

0

⋯

a

3

⁢

n

⋮

⋮

⋮

⋱

⋮

0

0

0

⋯

0

]

A strictly lower triangular matrix is of the form

[

0

0

0

⋯

0

a

21

0

0

⋯

0

a

31

a

32

0

⋯

0

⋮

⋮

⋮

⋱

⋮

a

n

⁢

1

a

n

⁢

2

a

n

⁢

3

⋯

0

]

Title

strictly upper triangular matrix

Canonical

name

StrictlyUpperTriangularMatrix

Date of creation

2013-03-22 13:42:15

Last modified on

2013-03-22 13:42:15

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

8

Author

Daume (40)

Entry type

Definition

Classification

msc 15-00

Synonym

supertriangular

Defines

strictly lower triangular matrix

Generated on Fri Feb 9 18:16:20 2018 by

LaTeXML
submatrix notation

submatrix notation

Let

n

and

k

be integers with

1

≤

k

≤

n

. Denote by

Q

k

,

n

the totality of all

sequences

of

k

integers, where the elements of
the sequence are

strictly increasing

and choosen from

{

1

,

…

,

n

}

.

Let

A

=

(

a

i

⁢

j

)

be an

m

×

n

matrix with elements from some set, usually
taken to be a field for ring. Let

k

and

r

be positive integers with

1

≤

k

≤

m

,

1

≤

r

≤

n

,

α

∈

Q

k

,

m

and

β

∈

Q

r

,

n

.
We let

α

=

(

i

1

,

…

,

i

k

)

and

β

=

(

j

1

,

…

,

j

r

)

The

submatrix

A

⁢

[

α

,

β

]

has

(

s

,

t

)

entry equal to

a

i

s

⁢

j

t

and has

k

rows and

r

columns.

We denote by

A

⁢

(

α

,

β

)

the submatrix of

A

whose rows and columns
are

complementary

to

α

and

β

, respectively.

We can also define similarly the notations

A

⁢

[

α

,

β

)

and

A

⁢

(

α

,

β

]

.

Title

submatrix notation

Canonical

name

SubmatrixNotation

Date of creation

2013-03-22 16:13:36

Last modified on

2013-03-22 16:13:36

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

5

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 15-00

Generated on Fri Feb 9 18:16:22 2018 by

LaTeXML
symmetric matrix

symmetric matrix

Definition

:

Let

A

=

(

a

i

⁢

j

)

be a

square matrix

of
order

n

. The matrix

A

is

symmetric

if

a

i

⁢

j

=

a

j

⁢

i

for all

1

≤

i

≤

n

,

1

≤

j

≤

n

.

A

=

(

a

11

⋯

a

1

⁢

n

⋮

⋱

⋮

a

n

⁢

1

⋯

a

n

⁢

n

)

Properties

:

1.

A

t

=

A

where

A

t

is the matrix transpose

Examples:

•

(

a

b

b

c

)

•

(

a

b

c

b

d

e

c

e

f

)

Title

symmetric matrix

Canonical

name

SymmetricMatrix

Date of creation

2013-03-22 12:00:58

Last modified on

2013-03-22 12:00:58

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

13

Author

Daume (40)

Entry type

Definition

Classification

msc 15-00

Synonym

symmetric

Related topic

SelfDual

Related topic

HessianMatrix

Related topic

SkewHermitianMatrix

Generated on Fri Feb 9 18:16:23 2018 by

LaTeXML
topological proof of the Cayley-Hamilton theorem

topological proof of the Cayley-Hamilton theorem

We begin by showing that the theorem is true if the

characteristic
polynomial

does not have repeated roots, and then prove the general case.

Suppose then that the

discriminant

of the characteristic polynomial is
non-zero, and hence that

T

:

V

→

V

has

n

=

dim

⁡

V

distinct

eigenvalues

once we extend

1

1

Technically, this means that we must
work with the

vector space

V

¯

=

V

⊗

k

¯

, where

k

¯

is the

algebraic closure

of the original field of scalars, and with

T

¯

:

V

¯

→

V

¯

the extended

automorphism

with action

T

¯

⁢

(

v

⊗

a

)

→

T

⁢

(

V

)

⊗

a

,

v

∈

V

,

a

∈

k

¯

.

to the algebraic closure of the ground
field.
We can therefore choose a basis of

eigenvectors

, call them

𝐯

1

,

…

,

𝐯

n

, with

λ

1

,

…

,

λ

n

the
corresponding eigenvalues. From the definition of characteristic
polynomial we have that

c

T

⁢

(

x

)

=

∏

i

=

1

n

(

x

-

λ

i

)

.

The factors on the right commute, and hence

c

T

⁢

(

T

)

⁢

𝐯

i

=

0

for all

i

=

1

,

…

,

n

. Since

c

T

⁢

(

T

)

annihilates a basis, it must, in
fact, be zero.

To prove the general case, let

δ

⁢

(

p

)

denote the discriminant of
a

polynomial

p

, and let us remark that the discriminant mapping

T

↦

δ

⁢

(

c

T

)

,

T

∈

End

⁢

(

V

)

is polynomial on

End

⁢

(

V

)

. Hence the set of

T

with distinct eigenvalues is a dense
open subset of

End

⁢

(

V

)

relative to the

Zariski
topology

. Now the characteristic polynomial map

T

↦

c

T

⁢

(

T

)

,

T

∈

End

⁢

(

V

)

is a polynomial map on the vector
space

End

⁢

(

V

)

. Since it vanishes on a dense
open subset, it must vanish identically. Q.E.D.

Title

topological proof of the

Cayley-Hamilton theorem

Canonical name

TopologicalProofOfTheCayleyHamiltonTheorem

Date of creation

2013-03-22 12:33:22

Last modified on

2013-03-22 12:33:22

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

7

Author

rmilson (146)

Entry type

Proof

Classification

msc 15-00

Generated on Fri Feb 9 18:16:25 2018 by

LaTeXML
tridiagonal matrix

tridiagonal matrix

An

n

×

n

tridiagonal

matrix is of the form

[

d

1

u

1

0

0

⋯

0

l

1

d

2

u

2

0

⋯

0

0

l

2

d

3

u

3

⋯

0

⋮

⋮

⋱

⋱

⋱

⋮

0

0

⋯

l

n

-

2

d

n

-

1

u

n

-

1

0

0

⋯

0

l

n

-

1

d

n

]

Title

tridiagonal matrix

Canonical

name

TridiagonalMatrix

Date of creation

2013-03-22 12:11:37

Last modified on

2013-03-22 12:11:37

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

5

Author

akrowne (2)

Entry type

Definition

Classification

msc 15-00

Classification

msc 65-00

Synonym

tridiagonal

Related topic

PentadiagonalMatrix

Generated on Fri Feb 9 18:16:27 2018 by

LaTeXML
under determined

under determined

An

under determined

system of linear equations

has more unknowns than equations. It can be

consistent

with infinitely many solutions, or have no solution.

Title

under determined

Canonical name

UnderDetermined

Date of creation

2013-03-22 12:07:03

Last modified on

2013-03-22 12:07:03

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

8

Author

akrowne (2)

Entry type

Definition

Classification

msc 15-00

Synonym

under-determined

Synonym

underdetermined

Related topic

Overdetermined

Related topic

RankNullityTheorem

Generated on Fri Feb 9 18:16:28 2018 by

LaTeXML
unit triangular matrix

unit triangular matrix

A

unit triangular matrix

is a

triangular matrix

with

1

on the

diagonal

.

i.e.

A

unit upper triangular matrix

is of the form

[

1

a

12

a

13

⋯

a

1

⁢

n

0

1

a

23

⋯

a

2

⁢

n

0

0

1

⋯

a

3

⁢

n

⋮

⋮

⋮

⋱

⋮

0

0

0

⋯

1

]

and is sometimes called a

unit right triangular matrix

.

A

unit lower triangular matrix

is of the form

[

1

0

0

⋯

0

a

21

1

0

⋯

0

a

31

a

32

1

⋯

0

⋮

⋮

⋮

⋱

⋮

a

n

⁢

1

a

n

⁢

2

a

n

⁢

3

⋯

1

]

and is sometimes called a

unit left triangular matrix

.

Title

unit triangular matrix

Canonical name

UnitTriangularMatrix

Date of creation

2013-03-22 13:41:58

Last modified on

2013-03-22 13:41:58

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

6

Author

Daume (40)

Entry type

Definition

Classification

msc 15-00

Synonym

unit left triangular matrix

Synonym

unit right triangular matrix

Defines

unit upper triangular matrix

Defines

unit lower triangular matrix

Generated on Fri Feb 9 18:16:30 2018 by

LaTeXML
vector subspace

vector subspace

Definition

Let

V

be a

vector space

over a field

F

,
and let

W

be a subset of

V

.
If

W

is itself a vector space,
then

W

is said to be a

vector subspace

of

V

.
If in addtition

V

≠

W

, then

W

is a

proper vector subspace

of

V

.

If

W

is a nonempty subset of

V

,
then a necessary and

sufficient

condition for

W

to be a

subspace

is that

a

+

γ

⁢

b

∈

W

for all

a

,

b

∈

W

and all

γ

∈

F

.

0.0.1

Examples

1.

Every vector space is a vector subspace of itself.

2.

In every vector space,

{

0

}

is a vector subspace.

3.

If

S

and

T

are vector subspaces of a vector space

V

,
then the

vector sum

S

+

T

=

{

s

+

t

∈

V

∣

s

∈

S

,

t

∈

T

}

and the

intersection

S

∩

T

=

{

u

∈

V

∣

u

∈

S

,

u

∈

T

}

are vector subspaces of

V

.

4.

Suppose

S

and

T

are vector spaces,
and suppose

L

is a

linear
mapping

L

:

S

→

T

.
Then

Im

⁡

L

is a vector subspace of

T

,
and

Ker

⁡

L

is a vector subspace of

S

.

5.

If

V

is an

inner product space

,
then the

orthogonal complement

of any subset of

V

is a vector subspace of

V

.

0.0.2

Results for vector subspaces

Theorem

1

[

1

]

Let

V

be a finite dimensional vector space.
If

W

is a vector subspace of

V

and

dim

⁡

W

=

dim

⁡

V

, then

W

=

V

.

Theorem 2

[

2

]

(Dimension theorem for subspaces)
Let

V

be a vector space with
subspaces

S

and

T

. Then

dim

⁡

(

S

+

T

)

+

dim

⁡

(

S

∩

T

)

=

dim

⁡

S

+

dim

⁡

T

.

References

1

S. Lang,

Linear Algebra

,
Addison-Wesley, 1966.

2

W.E. Deskins,

Abstract Algebra

,
Dover publications,
1995.

Title

vector subspace

Canonical name

VectorSubspace

Date of creation

2013-03-22 11:55:24

Last modified on

2013-03-22 11:55:24

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

20

Author

yark (2760)

Entry type

Definition

Classification

msc 15-00

Synonym

subspace

Synonym

linear subspace

Related topic

VectorSpace

Related topic

LinearManifold

Defines

dimension theorem for subspaces

Defines

proper vector subspace

Generated on Fri Feb 9 18:16:32 2018 by

LaTeXML
zero map

zero map

Definition

Suppose

X

is a set, and

Y

is a

vector space

with

zero vector

0

.
If

Z

is a map

Z

:

X

→

Y

, such that

Z

⁢

(

x

)

=

0

for all

x

in

X

,
then

Z

is a

zero map

.

0.0.1

Examples

1.

On the set of

non-invertible

n

×

n

matrices, the

determinant

is a zero map.

2.

If

X

is the

zero vector space

, any

linear map

T

:

X

→

Y

is
a zero map. In fact,

T

⁢

(

0

)

=

T

⁢

(

0

⋅

0

)

=

0

⁢

T

⁢

(

0

)

=

0

.

3.

If

X

=

Y

and its field is

ℝ

or

ℂ

, then the

spectrum

of

Z

is

{

0

}

.

Title

zero map

Canonical name

ZeroMap

Date of creation

2013-03-22 14:03:38

Last modified on

2013-03-22 14:03:38

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

6

Author

matte (1858)

Entry type

Definition

Classification

msc 15-00

Related topic

ZeroVectorSpace

Related topic

ConstantFunction

Related topic

IdentityMap

Defines

zero operator

Generated on Fri Feb 9 18:16:34 2018 by

LaTeXML
zero vector space

zero vector space

Definition

A

zero vector space

is a

vector space

that contains only one element, a

zero vector

.

0.0.1

Properties

1.

Every vector space has a zero vector space as a

vector subspace

.

2.

A vector space

X

is a zero vector space if and only if the

dimension

of

X

is
zero.

3.

Any

linear map

defined on a zero vector space is the

zero map

.
If

T

is linear on

{

0

}

, then

T

⁢

(

0

)

=

T

⁢

(

0

⋅

0

)

=

0

⁢

T

⁢

(

0

)

=

0

.

Title

zero vector space

Canonical

name

ZeroVectorSpace

Date of creation

2013-03-22 14:03:32

Last modified on

2013-03-22 14:03:32

Owner

drini (3)

Last modified by

drini (3)

Numerical id

6

Author

drini (3)

Entry type

Definition

Classification

msc 15-00

Synonym

trivial vector space

Related topic

ZeroRing

Related topic

ZeroMap

Generated on Fri Feb 9 18:16:35 2018 by

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circulant matrix

circulant matrix

A

square matrix

M

:

A

×

A

→

C

is said to be

g

-

circulant

for an integer

g

if each row other than the first is obtained
from the preceding row by shifting the elements cyclically g columns to the right (g¿0) or -g columns to the left (g ¡ 0).

That is, if

A

=

[

a

i

⁢

j

]

then

a

i

,

j

=

a

i

+

1

,

j

+

g

where the subscripts are computed modulo d.
A 1-circulant is commonly called a circulant
and a -1-circulant is called a back circulant.

More explicitly, a matrix of the form

[

M

1

M

2

M

3

…

M

d

M

d

M

1

M

2

…

M

d

-

1

M

d

-

1

M

d

M

1

…

M

d

-

2

⋮

⋮

⋮

⋱

⋮

M

2

M

3

M

4

…

M

1

]

is called circulant.

Because the Jordan decomposition (

http://planetmath.org/JordanCanonicalFormTheorem

) of a

circulant matrix

is rather simple, circulant matrices have some

interest

in connection with the approximation of

eigenvalues

of
more general matrices. In particular, they have become part of the
standard apparatus in the computerized analysis of signals and images.

Title

circulant matrix

Canonical

name

CirculantMatrix

Date of creation

2013-03-22 13:53:38

Last modified on

2013-03-22 13:53:38

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

9

Author

bwebste (988)

Entry type

Definition

Classification

msc 15-01

Classification

msc 15A99

Generated on Fri Feb 9 18:16:37 2018 by

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derivative of inverse matrix

derivative of inverse matrix

Theorem 1

.

Suppose

A

is a

square matrix

depending on a real

parameter

t

taking values in an open set

I

⊆

R

. Further, suppose all

component

functions

in

A

are

differentiable

, and

A

⁢

(

t

)

is

invertible

for all

t

. Then, in

I

, we have

d

⁢

A

-

1

d

⁢

t

=

-

A

-

1

⁢

d

⁢

A

d

⁢

t

⁢

A

-

1

,

where

d

d

⁢

t

is the

derivative

.

Proof.

Suppose

a

i

⁢

j

⁢

(

t

)

are the component functions for

A

,
and

a

j

⁢

k

⁢

(

t

)

are component functions for

A

-

1

⁢

(

t

)

. Then
for each

t

we have

∑

j

=

1

n

a

i

⁢

j

⁢

(

t

)

⁢

a

j

⁢

k

⁢

(

t

)

=

δ

i

k

where

n

is the order of

A

, and

δ

i

k

is the

Kronecker delta

symbol. Hence

∑

j

=

1

n

d

⁢

a

i

⁢

j

d

⁢

t

⁢

a

j

⁢

k

+

a

i

⁢

j

⁢

d

⁢

a

j

⁢

k

d

⁢

t

=

0

,

that is,

d

⁢

A

d

⁢

t

⁢

A

-

1

=

-

A

⁢

d

⁢

A

-

1

d

⁢

t

from which the claim follows.
∎

Title

derivative of inverse matrix

Canonical name

DerivativeOfInverseMatrix

Date of creation

2013-03-22 14:43:52

Last modified on

2013-03-22 14:43:52

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

7

Author

matte (1858)

Entry type

Theorem

Classification

msc 15-01

Generated on Fri Feb 9 18:16:39 2018 by

LaTeXML
derivative of matrix

derivative of matrix

Suppose

I

is an open set of

ℝ

, and for each

t

∈

I

,

A

⁢

(

t

)

is
an

n

×

m

matrix. If each element in

A

⁢

(

t

)

is a

differentiable function

of

t

, we say that

A

is a

differentiable

, and define the

derivative

of

A

componentwise. This derivative we shall write as

d

d

⁢

t

⁢

A

or

d

⁢

A

d

⁢

t

.

Properties

In the below we assume that all matrices are dependent on a parameter

t

and the matrices are differentiable with respect to

t

.

1.

For any

n

×

m

matrix

A

,

(

d

⁢

A

d

⁢

t

)

T

=

d

d

⁢

t

⁢

(

A

T

)

,

where

T

is the matrix transpose.

2.

If

A

⁢

(

t

)

,

B

⁢

(

t

)

are matrices such that

A

⁢

B

is defined, then

d

d

⁢

t

⁢

(

A

⁢

B

)

=

d

⁢

A

d

⁢

t

⁢

B

+

A

⁢

d

⁢

B

d

⁢

t

.

3.

When

A

⁢

(

t

)

is

invertible

,

d

d

⁢

t

⁢

(

A

-

1

)

=

-

A

-

1

⁢

d

⁢

A

d

⁢

t

⁢

A

-

1

.

4.

For a

square matrix

A

⁢

(

t

)

,

tr

⁡

(

d

⁢

A

d

⁢

t

)

=

d

d

⁢

t

⁢

tr

⁡

(

A

)

,

where

tr

is the

matrix trace

.

5.

If

A

⁢

(

t

)

,

B

⁢

(

t

)

are

n

×

m

matrices and

A

∘

B

is the

Hadamard product

of

A

and

B

, then

d

d

⁢

t

⁢

(

A

∘

B

)

=

d

⁢

A

d

⁢

t

∘

B

+

A

∘

d

⁢

B

d

⁢

t

.

Title

derivative of matrix

Canonical name

DerivativeOfMatrix

Date of creation

2013-03-22 15:00:28

Last modified on

2013-03-22 15:00:28

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

10

Author

matte (1858)

Entry type

Definition

Classification

msc 15-01

Related topic

NthDerivativeOfADeterminant

Generated on Fri Feb 9 18:16:41 2018 by

LaTeXML
direct sum of matrices

direct sum of matrices

Direct sum of matrices

Let

A

be an

m

×

n

matrix and

B

be a

p

×

q

matrix. By the

direct sum

of

A

and

B

, written

A

⊕

B

, we mean the

(

m

+

p

)

×

(

n

+

q

)

matrix of the form

(

A

O

O

B

)

where the

O

’s represent

zero matrices

. The

O

on the top right is an

m

×

q

matrix, while the

O

on the bottom left is

n

×

p

.

For example, if

A

=

(

3

-

1

2

5

)

and

B

=

(

1

2

4

0

-

7

8

)

, then

(

A

O

O

B

)

=

(

3

-

1

0

0

2

5

0

0

0

0

1

2

0

0

4

2

0

0

-

7

8

)

Remark

.
It is not hard to see that the

⊕

operation on matrices is

associative

:

(

A

⊕

B

)

⊕

C

=

A

⊕

(

B

⊕

C

)

,

because both sides lead to

(

A

O

O

O

B

O

O

O

C

)

In fact, we can inductively define the direct sum of

n

matrices unambiguously.

Direct sums of linear transformations

The

direct sum of matrices

is closely related to the direct sum of

vector spaces

and

linear transformations

. Let

A

and

B

be as above, over some field

k

. We may view

A

and

B

as linear transformations

T

A

:

k

n

→

k

m

and

T

B

:

k

q

→

k

p

using the

standard ordered bases

. Then

A

⊕

B

may be viewed as the linear transformation

T

A

⊕

B

:

k

n

+

q

→

k

m

+

p

using the standard ordered basis, such that

•

the restriction of

T

A

⊕

B

to the

subspace

k

n

(embedded in

k

n

+

q

) is

T

A

, and

•

the restriction of

T

A

⊕

B

to

k

q

is

T

B

.

The above suggests that we can define direct sums on linear transformations. Let

T

1

:

V

1

→

W

1

and

T

2

:

V

2

→

W

2

be linear transformations, where

V

i

and

W

j

are

finite dimensional

vector spaces over some field

k

such that

V

1

∩

V

2

=

0

. Then define

T

1

⊕

T

2

:

V

1

⊕

V

2

→

W

1

⊕

W

2

such that for any

v

∈

V

1

⊕

V

2

,

(

T

1

⊕

T

2

)

⁢

(

v

1

,

v

2

)

:=

(

T

1

⁢

(

v

1

)

,

T

2

⁢

(

v

2

)

)

where

v

i

∈

V

i

.
Based on this definition, it is not hard to see that

T

A

⊕

B

=

T

A

⊕

T

B

for any matrices

A

and

B

.

More generally, if

β

i

is an ordered basis for

V

i

, then

β

:=

β

1

∪

β

2

extending the linear orders on

β

i

, such that if

v

i

∈

β

1

and

v

j

∈

β

2

, then

v

i

<

v

j

is an ordered basis for

V

1

⊕

V

2

, and

[

T

1

⊕

T

2

]

β

=

[

T

1

]

β

1

⊕

[

T

2

]

β

2

.

Title

direct sum of matrices

Canonical name

DirectSumOfMatrices

Date of creation

2013-03-22 17:36:48

Last modified on

2013-03-22 17:36:48

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 15-01

Related topic

DirectSum

Generated on Fri Feb 9 18:16:50 2018 by

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