aarjaneiro/planetmath huggingface dataset:
matrix unit

matrix unit

A

matrix unit

is a matrix (over some ring with

1

) whose entries are all

0

except in one cell, where it is

1

.

For example, among the

3

×

2

matrices,

(

1

0

0

0

0

0

)

,

(

0

1

0

0

0

0

)

,

(

0

0

1

0

0

0

)

,

(

0

0

0

1

0

0

)

,

(

0

0

0

0

1

0

)

,

(

0

0

0

0

0

1

)

are the matrix units.

Let

A

and

B

be

m

×

n

and

p

×

q

matrices over

R

, and

U

i

⁢

j

an

n

×

p

matrix unit (over

R

). Then

1.

A

⁢

U

i

⁢

j

is the

m

×

p

matrix whose

j

th column is the

i

th column of

A

, and

0

everywhere else, and

2.

U

i

⁢

j

⁢

B

is the

n

×

q

matrix whose

i

th row is the

j

th row of

B

and

0

everywhere else.

Remarks

. Let

M

=

M

m

×

n

⁢

(

R

)

be the set of all

m

by

n

matrices with entries in a ring

R

(with

1

). Denote

U

i

⁢

j

the matrix unit in

M

whose cell

(

i

,

j

)

is

1

.

•

M

is a (left or right)

R

-module generated by the

m

×

n

matrix units.

•

When

m

=

n

,

M

has the structure of an algebra over

R

. The matrix units have the following properties:

(a)

U

i

⁢

j

⁢

U

k

⁢

ℓ

=

δ

j

⁢

k

⁢

U

i

⁢

ℓ

, and

(b)

U

11

+

⋯

+

U

n

⁢

n

=

I

n

,

where

δ

i

⁢

j

is the

Kronecker delta

and

I

n

is the

identity matrix

. Note that the

U

i

⁢

i

form a complete set of pairwise

orthogonal idempotents

, meaning

U

i

⁢

i

⁢

U

i

⁢

i

=

U

i

⁢

i

and

U

i

⁢

i

⁢

U

j

⁢

j

=

0

if

i

≠

j

.

•

In general, in a

matrix ring

S

(consisting of, say, all

n

×

n

matrices), any set of

n

matrices satisfying the two properties above is called a

full set of matrix units

of

S

.

•

For example, if

{

U

i

⁢

j

∣

1

≤

i

,

j

≤

2

}

is the set of

2

×

2

matrix units over

ℝ

, then for any

invertible matrix

T

,

{

T

⁢

U

i

⁢

j

⁢

T

-

1

∣

1

≤

i

,

j

≤

2

}

is a full set of matrix units.

•

If we embed

R

as a

subring

of

M

n

⁢

(

R

)

, then

R

is the

centralizer

of the matrix units of

M

n

⁢

(

R

)

, meaning that the only elements in

M

n

⁢

(

R

)

that commute with the matrix units are the elements in

R

.

References

1

T. Y. Lam,

Lectures on Modules and Rings

, Springer, New York, 1998.

Title

matrix unit

Canonical name

MatrixUnit

Date of creation

2013-03-22 18:30:35

Last modified on

2013-03-22 18:30:35

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 15A30

Classification

msc 16S50

Related topic

ElementaryMatrix

Defines

full set of matrix units

Generated on Fri Feb 9 18:27:26 2018 by

LaTeXML
example of permutation matrix

example of permutation matrix

Consider the matrix

P

=

(

0

1

0

0

0

0

0

1

1

0

0

0

0

0

1

0

)

that corresponds to permuting the columns of the

identity matrix

under the

permutation

(

1243

)

(i.e. (

http://planetmath.org/Ie

), the first column of the identity matrix is the second column of

P

, the second column of the identity matrix is the fourth column of

P

, etc.). Then

P

is a

permutation matrix

.

We will consider what happens when we multiply a

4

×

4

matrix by

P

. For example, let

A

be the matrix

A

=

(

4

2

6

8

1

3

5

7

1

0

1

0

-

1

-

2

-

3

-

4

)

.

Then

P

⁢

A

=

(

0

1

0

0

0

0

0

1

1

0

0

0

0

0

1

0

)

⁢

(

4

2

6

8

1

3

5

7

1

0

1

0

-

1

-

2

-

3

-

4

)

=

(

1

3

5

7

-

1

-

2

-

3

-

4

4

2

6

8

1

0

1

0

)

.

We notice that

P

⁢

A

has the same rows as

A

. Moreover, the rows of

A

are the rows of

P

⁢

A

permuted under

(

1243

)

: The first row of

P

⁢

A

is the second row of

A

, the second row of

P

⁢

A

is the fourth row of

A

, etc.

Title

example of permutation matrix

Canonical name

ExampleOfPermutationMatrix

Date of creation

2013-03-22 15:03:14

Last modified on

2013-03-22 15:03:14

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

8

Author

Wkbj79 (1863)

Entry type

Example

Classification

msc 15A36

Generated on Fri Feb 9 18:27:32 2018 by

LaTeXML
permutation matrix

permutation matrix

1

Permutation Matrix

Let

n

be a positive integer. A

permutation matrix

is any

n

×

n

matrix which can be created by rearranging the rows and/or columns of the

n

×

n

identity matrix

. More formally, given a

permutation

π

from the

symmetric group

S

n

, one can define an

n

×

n

permutation matrix

P

π

by

P

π

=

(

δ

i

⁢

π

⁢

(

j

)

)

, where

δ

denotes the

Kronecker delta

symbol.

Premultiplying an

n

×

n

matrix

A

by an

n

×

n

permutation matrix results in a rearrangement of the rows of

A

. For example, if the matrix

P

is obtained by swapping rows

i

and

j

of the

n

×

n

identity matrix, then rows

i

and

j

of

A

will be swapped in the product

P

⁢

A

.

Postmultiplying an

n

×

n

matrix

A

by an

n

×

n

permutation matrix results in a rearrangement of the columns of

A

. For example, if the matrix

P

is obtained by swapping rows

i

and

j

of the

n

×

n

identity matrix, then columns

i

and

j

of

A

will be swapped in the product

A

⁢

P

.

2

Properties

Permutation matrices have the following properties:

•

They are

orthogonal

(

http://planetmath.org/OrthogonalMatrices

).

•

They are

invertible

.

•

For a fixed (

http://planetmath.org/Fixed3

) positive integer

n

, the

n

×

n

permutation matrices form a group under

matrix multiplication

.

•

Since they have a single 1 in each row

and

each column, they are

doubly stochastic

.

•

They are the

extreme points

of the

convex set

of

doubly stochastic matrices

.

Title

permutation matrix

Canonical name

PermutationMatrix

Date of creation

2013-03-22 12:06:39

Last modified on

2013-03-22 12:06:39

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

20

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 15A36

Related topic

MonomialMatrix

Generated on Fri Feb 9 18:27:34 2018 by

LaTeXML
Farkas lemma

Farkas lemma

Given an

m

×

n

matrix

A

and an

1

×

n

real

row vector

c

, both with real

coefficients

, one and only one of the following systems has a solution:

1.

A

⁢

x

≤

0

and

c

⁢

x

>

0

for some

n

-

column vector

x

;

2.

w

⁢

A

=

c

and

w

≥

0

for some

m

-row vector

w

.

Equivalently, one and only one of the following has a solution:

1.

A

⁢

x

≤

0

,

x

≤

0

and

c

⁢

x

>

0

for some

n

-column vector

x

;

2.

w

⁢

A

≤

c

and

w

≥

0

for some

m

-row vector

w

.

Remark.

Here,

A

⁢

x

≥

0

means that every of

A

⁢

x

is nonnegative, and similarly with the other expressions.

Title

Farkas lemma

Canonical name

FarkasLemma

Date of creation

2013-03-22 13:47:37

Last modified on

2013-03-22 13:47:37

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

11

Author

Koro (127)

Entry type

Theorem

Classification

msc 15A39

Synonym

Farkas theorem

Generated on Fri Feb 9 18:27:36 2018 by

LaTeXML
Farkas lemma, proof of

Farkas lemma, proof of

We begin by showing that at least one of the systems
has a solution.

Suppose that system 2
has no solution. Let

S

be the cone in

ℝ

n

generated by nonnegative

linear combinations

of the rows

a

1

,

…

,

a

m

of

A

. The set

S

is closed and convex. Since system 2 is unsolvable, the vector

c

is not in

S

; therefore, there exist a scalar

α

and

n

-

column vector

x

such that the

hyperplane

x

T

⁢

v

=

α

separates

c

from

S

in

ℝ

n

. This hyperplane can be selected so that for any point

v

∈

S

,

x

T

⁢

c

T

>

α

>

x

T

⁢

v

T

.

Since

0

∈

S

, this implies that

α

>

0

. Hence for any

w

≥

0

,

α

>

x

T

⁢

(

w

⁢

A

)

T

=

w

⁢

A

⁢

x

=

∑

i

=

0

m

w

i

⁢

(

A

⁢

x

)

i

.

Each

(

A

⁢

x

)

i

is nonpositive. Otherwise, by selecting

w

with

w

i

sufficiently large and all other

w

j

=

0

, we would get a

contradiction

.
We have now shown that

x

satisfies

A

⁢

x

≤

0

and

c

⁢

x

=

(

c

⁢

x

)

T

=

x

T

⁢

c

T

>

α

>

0

, which means that

x

is a solution of system 1. Thus at least one of the systems is solvable.

We claim that systems 1 and 2 are not simultaneously solvable. Suppose
that

x

is a solution of system 1 and

w

is a solution of system 2. Then
for each

i

, the inequality

w

i

⁢

(

A

⁢

x

)

i

≤

0

holds, and so

w

⁢

(

A

⁢

x

)

≤

0

.
However,

(

w

⁢

A

)

⁢

x

=

c

⁢

x

>

0

,

a contradiction. This

completes

the proof.

Title

Farkas lemma

, proof of

Canonical name

FarkasLemmaProofOf

Date of creation

2013-03-22 14:12:51

Last modified on

2013-03-22 14:12:51

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Proof

Classification

msc 15A39

Generated on Fri Feb 9 18:27:38 2018 by

LaTeXML
Bauer-Fike theorem

Bauer-Fike theorem

Let

λ

~

be a

complex number

and

u

~

be a vector with

∥

u

~

∥

p

=

1

, and let

r

=

A

⁢

u

~

-

λ

~

⁢

u

~

(usually,

λ

~

and

u

~

are considered to be approximation of an

eigenvalue

and of an

eigenvector

of

A

). Assume

A

is

diagonalizable

and

A

=

X

⁢

D

⁢

X

-

1

, with

D

a

diagonal matrix

. Then the matrix

A

has an eigenvalue

λ

which satisfies the

inequality

:

|

λ

-

λ

~

|

≤

κ

p

⁢

(

X

)

⁢

∥

r

∥

p

see also:

•

Wikipedia,

http://en.wikipedia.org/wiki/Bauer-Fike_Theorem

Bauer-Fike Theorem

Title

Bauer-Fike theorem

Canonical name

BauerFikeTheorem

Date of creation

2013-03-22 14:48:31

Last modified on

2013-03-22 14:48:31

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

12

Author

Andrea Ambrosio (7332)

Entry type

Theorem

Classification

msc 15A42

Generated on Fri Feb 9 18:27:40 2018 by

LaTeXML
Gershgorin’s circle theorem

Gershgorin’s circle theorem

Let

A

be a square

complex matrix

. Around every element

a

i

⁢

i

on the

diagonal

of the matrix, we draw a circle with radius the sum of the norms of the other elements on the same row

∑

j

≠

i

|

a

i

⁢

j

|

. Such circles are called

Gershgorin discs

.

Theorem: Every

eigenvalue

of A lies in one of these Gershgorin discs.

Proof: Let

λ

be an eigenvalue of

A

and

x

its corresponding

eigenvector

. Choose

i

such that

|

x

i

|

=

max

j

⁡

|

x

j

|

. Since

x

can’t be

0

,

|

x

i

|

>

0

. Now

A

⁢

x

=

λ

⁢

x

, or looking at the

i

-th

component

(

λ

-

a

i

⁢

i

)

⁢

x

i

=

∑

j

≠

i

a

i

⁢

j

⁢

x

j

.

Taking the norm on both sides gives

|

λ

-

a

i

⁢

i

|

=

|

∑

j

≠

i

a

i

⁢

j

⁢

x

j

x

i

|

≤

∑

j

≠

i

|

a

i

⁢

j

|

.

Title

Gershgorin’s circle theorem

Canonical name

GershgorinsCircleTheorem

Date of creation

2013-03-22 13:14:15

Last modified on

2013-03-22 13:14:15

Owner

lieven (1075)

Last modified by

lieven (1075)

Numerical id

7

Author

lieven (1075)

Entry type

Theorem

Classification

msc 15A42

Synonym

Gershgorin’s disc theorem

Synonym

Gerschgorin’s circle theorem

Synonym

Gerschgorin’s disc theorem

Related topic

BrauersOvalsTheorem

Defines

Gershgorin disc

Defines

Gerschgorin disc

Generated on Fri Feb 9 18:27:46 2018 by

LaTeXML
Gershgorin’s circle theorem result

Gershgorin’s circle theorem result

Since the

eigenvalues

of

A

and

A

transpose

are the same, you can get an additional set of discs which has the same centers,

a

i

⁢

i

, but a radius calculated by the column

∑

j

≠

i

|

a

j

⁢

i

|

(instead of the rows). If a disc is isolated it must contain an eigenvalue. The eigenvalues must lie in the

intersection

of these circles. Hence, by comparing the row and column discs, the eigenvalues may be located efficiently.

Title

Gershgorin’s circle theorem result

Canonical name

GershgorinsCircleTheoremResult

Date of creation

2013-03-22 13:48:47

Last modified on

2013-03-22 13:48:47

Owner

saki (2816)

Last modified by

saki (2816)

Numerical id

11

Author

saki (2816)

Entry type

Result

Classification

msc 15A42

Generated on Fri Feb 9 18:27:47 2018 by

LaTeXML
minimal Gershgorin set

minimal Gershgorin set

related to Gershgorin’s theorem is the so called ”

minimal Gershgorin set

”.
For every

A

∈

𝐂

n

,

n

,

𝐱

>

0

meaning

x

i

>

0

∀

i

, let’s define its

minimal Gershgorin set

G

⁢

(

A

)

as:

G

⁢

(

A

)

=

⋂

𝐱

>

0

G

𝐱

⁢

(

A

)

,

where

G

𝐱

⁢

(

A

)

=

⋃

i

=

1

n

{

z

∈

𝐂

:

|

z

-

a

i

⁢

i

|

≤

1

x

i

⁢

∑

j

≠

i

|

a

i

⁢

j

|

⁢

x

j

}

.

Theorem: Let

A

∈

𝐂

n

,

n

, let

σ

⁢

(

A

)

be the

spectrum

of

A

and let

G

⁢

(

A

)

be its minimal Gershgorin set defined as above. Then

σ

⁢

(

A

)

⊆

G

⁢

(

A

)

.

Proof.

Given

𝐱

>

0

, let

X

=

d

⁢

i

⁢

a

⁢

g

⁢

{

x

1

,

x

2

,

…

,

x

n

}

and let

B

X

=

X

-

1

⁢

A

⁢

X

. Then

A

and

B

X

share the same spectrum, being

similar

. Due to definition, and keeping in mind that

X

-

1

=

d

⁢

i

⁢

a

⁢

g

⁢

{

x

1

-

1

,

x

2

-

1

,

…

,

x

n

-

1

}

, we have

b

i

⁢

j

(

X

)

=

a

i

⁢

j

⁢

x

j

x

i

and, applying Gershgorin theorem to

B

X

, we get:

σ

⁢

(

A

)

=

σ

⁢

(

B

X

)

⊆

⋃

i

=

1

n

{

z

∈

𝐂

:

|

z

-

a

i

⁢

i

|

≤

1

x

i

⁢

∑

j

≠

i

|

a

i

⁢

j

|

⁢

x

j

}

and, since this is true for any

𝐱

>

0

, we finally get the thesis.
∎

Title

minimal Gershgorin set

Canonical name

MinimalGershgorinSet

Date of creation

2013-03-22 15:35:57

Last modified on

2013-03-22 15:35:57

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

11

Author

Andrea Ambrosio (7332)

Entry type

Definition

Classification

msc 15A42

Generated on Fri Feb 9 18:27:50 2018 by

LaTeXML
Ostrowski theorem

Ostrowski theorem

Let

A

be a complex

n

×

n

matrix,

R

i

=

∑

j

≠

i

|

a

i

⁢

j

|

,

C

j

=

∑

i

≠

j

|

a

i

⁢

j

|

1

≤

i

≤

n

,

1

≤

j

≤

n

. Let’s consider, for any

α

∈

(

0

,

1

)

, the circles of this kind:

O

i

=

{

z

∈

𝐂

:

|

z

-

a

i

⁢

i

|

≤

R

i

α

⁢

C

i

1

-

α

}

1

≤

i

≤

n

.

Theorem

(A. Ostrowski): For any

α

∈

(

0

,

1

)

, all the

eigenvalues

of

A

lie in the union of these

n

circles:

σ

⁢

(

A

)

⊆

⋃

i

O

i

.

Proof.

If

R

i

=

0

, the theorem says

a

i

⁢

i

is an eigenvalue, which
is obviously true. Let’s then concentrate on the

R

i

≠

0

. By eigenvalue

definition

, we have:

(

λ

-

a

i

⁢

i

)

⁢

x

i

=

∑

j

≠

i

a

i

⁢

j

⁢

x

j

so that, recalling Hölder’s inequality with

p

=

1

/

α

and

q

=

1

/

(

1

-

α

)

(to have

p

,

q

>

1

, we must have

α

∈

(

0

,

1

)

)

|

λ

-

a

i

⁢

i

|

⁢

|

x

i

|

≤

∑

j

≠

i

|

a

i

⁢

j

|

⁢

|

x

j

|

=

∑

j

≠

i

|

a

i

⁢

j

|

α

⁢

|

a

i

⁢

j

|

1

-

α

⁢

|

x

j

|

≤

(

∑

j

≠

i

(

|

a

i

⁢

j

|

α

)

1

/

α

)

α

⁢

(

∑

j

≠

i

(

|

a

i

⁢

j

|

1

-

α

⁢

|

x

j

|

)

1

/

(

1

-

α

)

)

1

-

α

=

(

∑

j

≠

i

|

a

i

⁢

j

|

)

α

⁢

(

∑

j

≠

i

|

a

i

⁢

j

|

⁢

|

x

j

|

1

/

(

1

-

α

)

)

1

-

α

=

R

i

α

⁢

(

∑

j

≠

i

|

a

i

⁢

j

|

⁢

|

x

j

|

1

/

(

1

-

α

)

)

1

-

α

which means

|

λ

-

a

i

⁢

i

|

1

/

(

1

-

α

)

R

i

α

/

(

1

-

α

)

⁢

|

x

i

|

1

/

(

1

-

α

)

≤

∑

j

≠

i

|

a

i

⁢

j

|

⁢

|

x

j

|

1

/

(

1

-

α

)

Summing

over all

i

, one obtains

∑

i

=

1

n

|

λ

-

a

i

⁢

i

|

1

/

(

1

-

α

)

R

i

α

/

(

1

-

α

)

⁢

|

x

i

|

1

/

(

1

-

α

)

≤

∑

i

=

1

n

∑

j

≠

i

|

a

i

⁢

j

|

⁢

|

x

j

|

1

/

(

1

-

α

)

=

∑

j

=

1

n

C

j

⁢

|

x

j

|

1

/

(

1

-

α

)

If, for each

i

, the coefficient of

|

x

i

|

1

/

(

1

-

α

)

in the first
sum would be greater than the coefficient of the same term in the right-hand
side, inequality couldn’t hold. So we can conclude that at least one index

p

exists
such as

|

λ

-

a

p

⁢

p

|

1

/

(

1

-

α

)

R

p

α

/

(

1

-

α

)

≤

C

p

that is

|

λ

-

a

p

⁢

p

|

≤

R

p

α

⁢

C

p

1

-

α

which is the thesis.
∎

Remarks:

The Gershgorin theorem is obtained as a limit for

α

→

0

or
for

α

→

1

; in other words, Ostrowski’s theorem represents a kind of ”

continuous deformation

” between the two Gershgorin rows and columns sets.

References

1

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

Matrix Analysis

, Cambridge University Press, 1985

Title

Ostrowski theorem

Canonical

name

OstrowskiTheorem

Date of creation

2013-03-22 15:36:29

Last modified on

2013-03-22 15:36:29

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

22

Author

Andrea Ambrosio (7332)

Entry type

Theorem

Classification

msc 15A42

Generated on Fri Feb 9 18:27:55 2018 by

LaTeXML
proof of Bauer-Fike theorem

proof of Bauer-Fike theorem

We can assume

λ

~

∉

σ

⁢

(

A

)

(otherwise, we can choose

λ

=

λ

~

and theorem is proven, since

κ

p

⁢

(

X

)

>

1

).
Then

(

A

-

λ

~

⁢

I

)

-

1

exists, so we can write:

u

~

=

(

A

-

λ

~

⁢

I

)

-

1

⁢

r

=

X

⁢

(

D

-

λ

~

⁢

I

)

-

1

⁢

X

-

1

⁢

r

since

A

is

diagonalizable

; taking the p-norm (

http://planetmath.org/VectorPnorm

) of both sides, we obtain:

1

=

∥

u

~

∥

p

=

∥

X

⁢

(

D

-

λ

~

⁢

I

)

-

1

⁢

X

-

1

⁢

r

∥

p

≤

∥

X

∥

p

⁢

∥

(

D

-

λ

~

⁢

I

)

-

1

∥

p

⁢

∥

X

-

1

∥

p

⁢

∥

r

∥

p

=

κ

p

⁢

(

X

)

⁢

∥

(

D

-

λ

~

⁢

I

)

-

1

∥

p

⁢

∥

r

∥

p

.

But, since

(

D

-

λ

~

⁢

I

)

-

1

is a

diagonal matrix

, the p-norm is easily computed, and yields:

∥

(

D

-

λ

~

⁢

I

)

-

1

∥

p

=

max

∥

x

∥

p

≠

0

⁡

∥

(

D

-

λ

~

⁢

I

)

-

1

⁢

x

∥

p

∥

x

∥

p

=

max

λ

∈

σ

⁢

(

A

)

⁡

1

|

λ

-

λ

~

|

=

1

min

λ

∈

σ

⁢

(

A

)

⁡

|

λ

-

λ

~

|

whence:

min

λ

∈

σ

⁢

(

A

)

⁡

|

λ

-

λ

~

|

≤

κ

p

⁢

(

X

)

⁢

||

r

||

p

.

Title

proof of

Bauer-Fike theorem

Canonical name

ProofOfBauerFikeTheorem

Date of creation

2013-03-22 15:33:08

Last modified on

2013-03-22 15:33:08

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

9

Author

Andrea Ambrosio (7332)

Entry type

Proof

Classification

msc 15A42

Generated on Fri Feb 9 18:27:59 2018 by

LaTeXML
proof of Weyl’s inequality

proof of Weyl’s inequality

Let

λ

i

be the i-th

eigenvalue

of

A

+

E

. Then, by the Courant-Fisher min-max theorem and being

x

H

⁢

E

⁢

x

≥

0

by hypothesis, we have:

λ

i

⁢

(

A

+

E

)

=

max

S

,

d

⁢

i

⁢

m

⁢

S

=

i

⁡

min

∥

x

∥

≠

0

⁡

x

H

⁢

(

A

+

E

)

⁢

x

x

H

⁢

x

=

=

max

S

,

dim

⁡

S

=

i

⁡

min

∥

x

∥

≠

0

⁡

(

x

H

⁢

A

⁢

x

x

H

⁢

x

+

x

H

⁢

E

⁢

x

x

H

⁢

x

)

≥

max

S

,

dim

⁡

S

=

i

⁡

min

∥

x

∥

≠

0

⁡

x

H

⁢

A

⁢

x

x

H

⁢

x

=

λ

i

⁢

(

A

)

.

Title

proof of Weyl’s inequality

Canonical name

ProofOfWeylsInequality

Date of creation

2013-03-22 15:33:39

Last modified on

2013-03-22 15:33:39

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

9

Author

Andrea Ambrosio (7332)

Entry type

Proof

Classification

msc 15A42

Generated on Fri Feb 9 18:28:01 2018 by

LaTeXML
Weyl’s inequality

Weyl’s inequality

Let

A

and

E

be two

n

×

n

Hermitian matrices

, with

E

positive semidefinite

.

Let

λ

i

⁢

(

A

)

,

λ

i

⁢

(

A

+

E

)

,

1

≤

i

≤

n

be the

eigenvalues

of

A

and

A

+

E

respectively, ordered in such a way that

|

λ

1

|

≤

|

λ

2

|

≤

⋯

≤

|

λ

n

|

.

Then

λ

i

⁢

(

A

)

≤

λ

i

⁢

(

A

+

E

)

.

Title

Weyl’s

inequality

Canonical name

WeylsInequality

Date of creation

2013-03-22 15:33:37

Last modified on

2013-03-22 15:33:37

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

10

Author

Andrea Ambrosio (7332)

Entry type

Theorem

Classification

msc 15A42

Generated on Fri Feb 9 18:28:03 2018 by

LaTeXML
Wielandt-Hoffman theorem

Wielandt-Hoffman theorem

Let

A

and

B

be

normal matrices

(

http://planetmath.org/NormalMatrix

). Let their

eigenvalues

a

i

and

b

i

be ordered such that

∑

i

|

a

i

-

b

i

|

2

is
minimized. Then we have the following inequality

∑

i

|

a

i

-

b

i

|

2

≤

∥

A

-

B

∥

F

2

,

where

∥

⋅

∥

F

is the

Frobenius matrix norm

.

Title

Wielandt-Hoffman theorem

Canonical name

WielandtHoffmanTheorem

Date of creation

2013-03-22 14:58:45

Last modified on

2013-03-22 14:58:45

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

4

Author

Andrea Ambrosio (7332)

Entry type

Theorem

Classification

msc 15A42

Classification

msc 15A18

Related topic

ShursInequality

Generated on Fri Feb 9 18:28:05 2018 by

LaTeXML
Hadamard’s inequality

Hadamard’s inequality

Let

A

=

(

a

i

⁢

j

)

with

1

≤

i

,

j

≤

n

∈

ℕ

be a

square matrix

with

complex

coefficients

. Then the following

inequality

holds:

|

det

⁡

(

A

)

|

≤

∏

i

=

1

n

(

∑

j

=

1

n

|

a

i

⁢

j

|

2

)

1

2

.

Moreover, if

A

is

Hermitian

and

positive semidefinite

, the following inequality holds:

det

⁡

(

A

)

≤

∏

i

=

1

n

a

i

⁢

i

,

with equality if and only if

A

is a

diagonal matrix

.

Title

Hadamard

’s inequality

Canonical name

HadamardsInequality

Date of creation

2013-03-22 14:32:21

Last modified on

2013-03-22 14:32:21

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

9

Author

mathwizard (128)

Entry type

Theorem

Classification

msc 15A45

Generated on Fri Feb 9 18:28:07 2018 by

LaTeXML
copositive matrix

copositive matrix

A

copositive matrix

is a real

symmetric matrix

M

such that for every real

n-vector

x

with all nonnegative

components

,

x

T

⁢

M

⁢

x

≥

0

.

The set of copositive matrices is a proper cone.

References

1

S. Boyd, L. Vandenberghe,

Convex Optimization

, Cambridge University Press, 2004.

Title

copositive matrix

Canonical name

CopositiveMatrix

Date of creation

2013-03-22 14:37:07

Last modified on

2013-03-22 14:37:07

Owner

dooder0001 (4288)

Last modified by

dooder0001 (4288)

Numerical id

6

Author

dooder0001 (4288)

Entry type

Definition

Classification

msc 15A48

Generated on Fri Feb 9 18:28:13 2018 by

LaTeXML
Euclidean distance matrix

Euclidean distance matrix

A

Euclidean distance matrix

(EDM) is a real

m

×

m

matrix

X

such that for some points

y

1

,

…

,

y

m

in

ℝ

m

,

X

i

⁢

k

=

∥

y

i

-

y

k

∥

2

2

, where

∥

⋅

∥

2

is the 2-norm on

ℝ

m

.

A EDM

X

inherits the following from the norm that defines it:

•

X

i

⁢

i

=

0

;

•

X

i

⁢

j

=

X

j

⁢

i

≥

0

;

•

X

i

⁢

k

≤

X

i

⁢

j

+

X

j

⁢

k

.

Additionally,

X

is a EDM if and only if the

diagonal

entries of

X

are all 0 and for all

z

∈

ℝ

m

whose

components

sum to 0,

z

T

⁢

X

⁢

z

≤

0

.

Finally, the set of

m

×

m

EDMs forms a convex cone (

http://planetmath.org/Cone3

) in the set of all

m

×

m

matrices.

References

1

S. Boyd, L. Vandenberghe,

Convex Optimization

, Cambridge University Press, 2004.

Title

Euclidean distance matrix

Canonical

name

EuclideanDistanceMatrix

Date of creation

2013-03-22 14:37:15

Last modified on

2013-03-22 14:37:15

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 15A48

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

LaTeXML