aarjaneiro/planetmath huggingface dataset:
orthogonal Latin squares

orthogonal Latin squares

Given two

Latin squares

L

1

=

(

A

,

B

,

C

1

,

f

1

)

and

L

2

=

(

A

,

B

,

C

2

,

f

2

)

of the same order

n

, we can combine them coordinate-wise to form a single square, whose cells are

ordered pairs

of

elements

from

C

1

and

C

2

respectively. Formally, we can form a

function

f

:

A

×

B

→

C

1

×

C

2

given by

f

⁢

(

i

,

j

)

=

(

f

1

⁢

(

i

,

j

)

,

f

2

⁢

(

i

,

j

)

)

.

This function

f

says that we have created a new square

A

×

B

, whose cell

(

i

,

j

)

contains the ordered pair of values, the first

coordinate

of which corresponds to the value in cell

(

i

,

j

)

of

L

1

, and the second to the value in cell

(

i

,

j

)

of

L

2

. We may write the combined square

L

1

*

L

2

.

For example,

(

a

b

c

d

c

d

a

b

d

c

b

a

b

a

d

c

)

*

(

1

2

3

4

4

3

2

1

2

1

4

3

3

4

1

2

)

=

(

(

a

,

1

)

(

b

,

2

)

(

c

,

3

)

(

d

,

4

)

(

c

,

4

)

(

d

,

3

)

(

a

,

2

)

(

b

,

1

)

(

d

,

2

)

(

c

,

1

)

(

b

,

4

)

(

a

,

3

)

(

b

,

3

)

(

a

,

4

)

(

d

,

1

)

(

c

,

2

)

)

In general, the combined square is not a Latin square unless the original two squares are

equivalent

:

f

1

⁢

(

i

,

j

)

=

f

1

⁢

(

k

,

ℓ

)

iff

f

2

⁢

(

i

,

j

)

=

f

2

⁢

(

k

,

ℓ

)

. Nevertheless, the more interesting aspect of

pairing

up two Latin squares (of the same order) lies in the function

f

:

Definition

. We say that two Latin squares are

orthogonal

if

f

is a

bijection

.

Since there are

n

2

cells in the combined square, and

|

C

1

×

C

2

|

=

n

2

, the function

f

is a bijection if it is either

one-to-one

or onto. It is therefore easy to see that the two Latin squares in the example above are orthogonal.

Remarks

.

•

The combined square is usually known as a

Graeco-Latin square

, originated from statisticians Fischer and Yates.

•

It can be shown that if

L

1

,

…

,

L

m

are Latin squares of order

n

≥

3

such that each pair of them are orthogonal, then

m

≤

n

-

1

. If the

equality

occurs, then the set of Latin squares are said to be

complete

.

•

(Bose) If

n

≥

3

, then

L

1

,

…

,

L

m

form a

complete set

of

pairwise orthogonal Latin squares

of order

n

iff there exists a

finite projective plane

of order

n

.

References

1

H. J. Ryser,

Combinatorial Mathematics

, The Carus Mathematical Monographs, 1963

Title

orthogonal Latin squares

Canonical name

OrthogonalLatinSquares

Date of creation

2013-03-22 16:04:47

Last modified on

2013-03-22 16:04:47

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 62K10

Classification

msc 05B15

Synonym

mutually orthogonal Latin squares

Synonym

MOLS

Synonym

pairwise orthogonal Latin squares

Defines

complete set of Latin squares

Generated on Sat Feb 10 12:06:12 2018 by

LaTeXML
Lévy process

Lévy process

In probability theory, a Lévy process, named after the French mathematician Paul Pierre Lévy is any continuous-time stochastic process that starts at 0, admits càdlàg (right-continuous with left limits)

modification

and has “

stationary independent increments

”. The most well-known examples are the

Wiener process

and the

Poisson process

.

A continuous-time stochastic process assigns a

random variable

X

t

to each point

t

≥

0

in time. In effect it is a

random function

of

t

. The increments of such a process are the

differences

X

s

-

X

t

between its values at different times

t

<

s

. To call the increments of a process

independent

means that increments

X

s

-

X

t

and

X

u

-

X

v

are independent random variables whenever the two time intervals do not overlap and, more generally, any finite number of increments assigned to pairwise non-overlapping time intervals are mutually (not just pairwise) independent. To call the increments

stationary

means that the probability distribution of any increment

X

s

-

X

t

depends only on the length

s

-

t

of the time interval; increments with equally long time intervals are

identically distributed

.

In the Wiener process, the probability distribution of

X

s

-

X

t

is normal with

expected value

0 and

variance

s

-

t

.

In the Poisson process, the probability distribution of

X

s

-

X

t

is a

Poisson distribution

with expected value

λ

⁢

(

s

-

t

)

, where

λ

>

0

is the intensity or rate of the process.

The probability distributions of the increments of any Lévy process are

infinitely divisible

. There is a Lévy process for each infinitely divisible probability distribution.

In any Lévy process with finite moments, the

n

th moment

μ

n

⁢

(

t

)

=

E

⁢

(

X

t

n

)

is a polynomial function of

t

; these

functions

satisfy

a

binomial identity

:

μ

n

⁢

(

t

+

s

)

=

∑

k

=

0

n

(

n

k

)

⁢

μ

k

⁢

(

t

)

⁢

μ

n

-

k

⁢

(

s

)

.

It is possible to characterise all Lévy processes by looking at their

characteristic function

.

This entry was

adapted

from the Wikipedia article

http://en.wikipedia.org/wiki/Levy_process

Lévy process as of January 25, 2007.

Title

Lévy process

Canonical

name

LevyProcess

Date of creation

2013-03-22 16:37:19

Last modified on

2013-03-22 16:37:19

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 62M09

Synonym

Levy process

Generated on Sat Feb 10 12:06:14 2018 by

LaTeXML
moving average

moving average

A

moving average

is a

sequence

of

arithmetic means

taken over a fixed

interval

moved along consecutive data points from an infinite (or sufficiently large) set of data points. That is, given a sequence

a

x

,

…

,

a

x

+

k

and an interval

n

, the

average

a

i

-

n

2

+

…

+

a

i

+

n

2

n

is taken for each value

(

x

+

n

)

<

i

<

(

x

+

k

-

n

)

.

Plotting a moving average can help to smooth out an extremely jagged curve so as to allow one to see larger patterns. For example, take this plot of the number of (nondistinct)

prime factors

function (

http://planetmath.org/NumberOfNondistinctPrimeFactorsFunction

)

Ω

⁢

(

n

)

for

20

<

n

<

120

:

If instead we plot a moving average with an interval of 40, we get a smoother curve:

Though in all honesty, moving averages are not all that useful in

number theory

. They are much used, however, in

statistics

and fields using statistics, such as physics and economics. In economics, for example, a moving average over an interval of say, 3 months, helps investors worry less about the wild hectic fluctuations in a day of trading and focus on the overall direction of a given stock. In physics, to give another example, a yearly moving average of parts per million of carbon dioxide in the atmosphere of the Earth smooths out the yearly dips of summer to show that overall carbon dioxide is going up, contributing to significant global warming.

Title

moving average

Canonical name

MovingAverage

Date of creation

2013-03-22 16:45:03

Last modified on

2013-03-22 16:45:03

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

9

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 62M10

Classification

msc 26D15

Classification

msc 11-00

Classification

msc 91B84

Generated on Sat Feb 10 12:06:16 2018 by

LaTeXML
hazard function

hazard function

Let

Y

be a

random variable

with

probability density function

f

Y

⁢

(

y

)

. Then the

hazard function

h

⁢

(

y

)

is defined to be:

h

⁢

(

y

)

=

f

Y

⁢

(

y

)

1

-

F

Y

⁢

(

y

)

=

f

Y

⁢

(

y

)

S

⁢

(

y

)

,

where

S

⁢

(

y

)

is the

survivor function

and

Y

is the survival time.

The hazard function is the rate of

probability

of death (non survival) is changing at time

Y

=

y

, given survival up to time

y

:

h

⁢

(

y

)

=

lim

Δ

⁢

y

→

0

⁡

P

(

y

≤

Y

≤

y

+

Δ

y

∣

Y

>

y

)

Δ

⁢

y

.

The

cumulative hazard function

,

H

⁢

(

y

)

of

Y

is defined as

H

⁢

(

y

)

=

∫

-

∞

y

h

⁢

(

t

)

⁢

𝑑

t

.

From this definition, we see that

H

⁢

(

y

)

=

-

ln

⁡

S

⁢

(

y

)

.

Examples

.
The hazard functions for the three most widely used probability density functions for survival time are:

•

The

exponential distribution

, with

h

⁢

(

y

)

=

γ

.

•

The

Weibull distribution

, with

h

⁢

(

y

)

=

γ

⁢

y

γ

-

1

using the standard Weibull distribution.

•

The extreme-value

distribution

, with

h

⁢

(

y

)

=

1

β

⁢

exp

⁡

(

y

-

α

β

)

.

Title

hazard function

Canonical name

HazardFunction

Date of creation

2013-03-22 14:27:45

Last modified on

2013-03-22 14:27:45

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 62N99

Classification

msc 62P05

Defines

cumulative hazard function

Generated on Sat Feb 10 12:06:17 2018 by

LaTeXML
survivor function

survivor function

Let

Y

be a

random variable

with cumulative

probability distribution function

F

Y

⁢

(

y

)

. Then the

survivor function

S

⁢

(

y

)

is defined to be:

S

(

y

)

=

1

-

F

Y

(

y

)

=

P

(

Y

≥

y

)

.

The random variable

Y

is often called the

survival time

.

The survivor function is the

probability

of survival beyond time

Y

=

y

.

Examples.

The three most commonly used

distribution functions

for survival time are:

1.

exponential distribution

(

http://planetmath.org/ExponentialRandomVariable

), with

S

⁢

(

y

)

=

exp

⁡

(

-

γ

⁢

y

)

.

2.

Weibull distribution

, with

S

⁢

(

y

)

=

exp

⁡

(

-

y

γ

)

using the standard Weibull distribution.

3.

extreme-value distribution, with

S

⁢

(

y

)

=

exp

⁡

(

-

exp

⁡

(

y

-

α

β

)

)

.

Title

survivor function

Canonical

name

SurvivorFunction

Date of creation

2013-03-22 14:27:43

Last modified on

2013-03-22 14:27:43

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 62N99

Classification

msc 62P05

Defines

survival time

Generated on Sat Feb 10 12:06:19 2018 by

LaTeXML
Weibull random variable

Weibull random variable

X

is a

Weibull random variable

if it has a

probability density function

, given by

f

X

⁢

(

x

)

=

γ

α

⁢

(

x

-

μ

α

)

γ

-

1

⁢

e

-

(

x

-

μ

α

)

γ

where

α

,

γ

,

μ

∈

ℝ

,

α

,

γ

>

0

and

x

≥

μ

.

α

is the

scale parameter

,

γ

is the

shape parameter

, and

μ

is the

location parameter

.

Notation for

X

having a

Weibull distribution

is

X

∼

Wei

⁢

(

α

,

γ

,

μ

)

. Usually, the location and scale parameters are dropped by the transformation

Y

=

X

-

μ

α

so that

Y

∼

Wei

⁢

(

γ

)

:=

Wei

⁢

(

1

,

γ

,

0

)

. The resulting

distribution

is called the

standard Weibull

, or

Rayleigh distribution

:

f

X

⁢

(

x

)

=

γ

⁢

x

γ

-

1

⁢

exp

⁡

(

-

x

γ

)

:
Given a standard Weibull distribution

X

∼

Wei

⁢

(

γ

)

:

1.

E

⁡

[

X

]

=

Γ

⁢

(

γ

+

1

γ

)

, where

Γ

is the

gamma function

2.

Median =

(

ln

⁡

2

)

1

γ

3.

Mode

=

{

(

1

-

1

γ

)

1

/

γ

if

γ

>

1

0

otherwise

4.

Var

⁡

[

X

]

=

Γ

⁢

(

γ

+

2

γ

)

-

Γ

⁢

(

γ

+

1

γ

)

2

5.

X

∼

Wei

⁢

(

α

,

γ

,

0

)

iff

X

γ

∼

Exp

⁢

(

α

γ

)

, the

exponential distribution

with

parameter

α

γ

Remark.

The Weibull distribution is often used to model reliability or lifetime of such as light bulbs.

Title

Weibull random variable

Canonical

name

WeibullRandomVariable

Date of creation

2013-03-22 14:26:44

Last modified on

2013-03-22 14:26:44

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 62N99

Classification

msc 62E15

Classification

msc 60E05

Classification

msc 62P05

Synonym

Weibull distribution

Synonym

Rayleigh distribution

Generated on Sat Feb 10 12:06:21 2018 by

LaTeXML
law of rare events

law of rare events

Let

X

be distributed as

B

⁢

i

⁢

n

⁢

(

n

,

p

)

, a binomial

random variable

with

parameters

n

and

p

. Suppose

lim

n

→

∞

⁡

n

⁢

p

=

λ

,

where

λ

is a

positive

real constant, then

X

is asymptotically distributed as

P

⁢

o

⁢

i

⁢

s

⁢

s

⁢

o

⁢

n

⁢

(

λ

)

, a

Poisson distribution

with parameter

λ

.

Basically, when the size of the

population

n

is very large and the occurrence of certain

event

A

is rare, where

p

, the

probability

of

A

is very small, the binomial random variable

X

can be approximated by a Poisson random variable.

Sketch of Proof.

Let

X

∼

B

⁢

i

⁢

n

⁢

(

n

,

p

)

. So

P

(

X

=

m

)

=

n

!

m

!

⁢

(

n

-

m

)

!

⁢

p

m

⁢

(

1

-

p

)

n

-

m

=

n

!

n

m

⁢

(

n

-

m

)

!

⁢

(

n

⁢

p

)

m

m

!

⁢

(

1

-

n

⁢

p

n

)

n

-

m

=

n

!

n

m

⁢

(

n

-

m

)

!

⁢

(

n

⁢

p

)

m

m

!

⁢

(

1

-

n

⁢

p

n

)

n

⁢

(

1

-

n

⁢

p

n

)

-

m

.

As

n

→

∞

,

n

!

n

m

⁢

(

n

-

m

)

!

=

n

n

⁢

n

-

1

n

⁢

⋯

⁢

n

-

m

+

1

n

≈

1

,

(

1

-

n

⁢

p

n

)

-

m

≈

(

1

-

λ

n

)

-

m

≈

1

,

(

1

-

n

⁢

p

n

)

n

≈

(

1

-

λ

n

)

n

≈

e

-

λ

,

and

(

n

⁢

p

)

m

m

!

≈

λ

m

m

!

.

Therefore,

P

(

X

=

m

)

≈

λ

m

m

!

e

-

λ

=

P

o

i

s

s

o

n

(

λ

)

.

Example.

Suppose in a given year, the number of fatal automobile accidents has a

binomial distribution

for a particular insuarance company with five

hundred

automobile insurance policies. On average, there is one policy out of the five hundred that will be involved in a fatal crash. What is the probability that there will be no fatal accidents (out of five hundred policies) in any particular year?

Solution.

If

X

be the number of fatal accidents in a year from a population of 500 auto policies, then

X

∼

B

⁢

i

⁢

n

⁢

(

n

,

p

)

with

n

=

500

and

p

=

1

/

500

.

λ

=

500

×

1

/

500

=

1

and so

P

(

X

=

0

)

≈

e

-

1

≈

0.368

.

Using the binomial distribution, we have

P

(

X

=

0

)

=

(

1

-

1

500

)

500

≈

0.367

.

Title

law of rare events

Canonical

name

LawOfRareEvents

Date of creation

2013-03-22 14:39:32

Last modified on

2013-03-22 14:39:32

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 62P05

Classification

msc 60E99

Classification

msc 60F99

Synonym

Poisson theorem

Generated on Sat Feb 10 12:06:23 2018 by

LaTeXML
Hessenberg matrix

Hessenberg matrix

An

upper Hessenberg matrix

is of the form

[

a

11

a

12

a

13

⋯

a

1

,

n

-

1

a

1

⁢

n

a

21

a

22

a

23

⋯

a

2

,

n

-

1

a

2

⁢

n

0

a

32

a

33

⋯

a

3

,

n

-

1

a

3

⁢

n

0

0

a

43

⋯

a

4

,

n

-

1

a

4

⁢

n

⋮

⋮

⋮

⋱

⋮

⋮

0

0

0

0

a

n

,

n

-

1

a

n

⁢

n

]

and a

lower Hessenberg matrix

is of the form

[

a

11

a

12

0

⋯

0

0

a

21

a

22

a

23

⋯

0

0

⋮

⋮

⋮

⋱

⋮

⋮

a

n

-

2

,

1

a

n

-

2

,

2

a

n

-

2

,

3

⋯

a

n

-

2

,

n

-

1

0

a

n

-

1

,

1

a

n

-

1

,

2

a

n

-

1

,

3

⋯

a

n

-

1

,

n

-

1

a

n

-

1

,

n

a

n

,

1

a

n

,

2

a

n

,

3

⋯

a

n

,

n

-

1

a

n

,

n

]

Title

Hessenberg matrix

Canonical name

HessenbergMatrix

Date of creation

2013-03-22 12:11:35

Last modified on

2013-03-22 12:11:35

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

6

Author

akrowne (2)

Entry type

Definition

Classification

msc 65-00

Classification

msc 15-00

Synonym

upper Hessenberg matrix

Synonym

lower Hessenberg matrix

Synonym

upper Hessenberg

Synonym

lower Hessenberg

Generated on Sat Feb 10 12:34:39 2018 by

LaTeXML
normal equations

normal equations

Normal Equations

We consider the problem

A

⁢

x

≈

b

, where

A

is an

m

×

n

matrix with

m

≥

n

rank

(

A

)

=

n

,

b

is an

m

×

1

vector, and

x

is the

n

×

1

vector to be determined.

The sign

≈

stands for the least squares approximation, i.e. a minimization of the norm of the

residual

r

=

A

⁢

x

-

b

.

||

A

⁢

x

-

b

||

2

=

||

r

||

2

=

[

∑

i

=

1

m

r

i

2

]

1

/

2

or the square

F

⁢

(

x

)

=

1

2

⁢

||

A

⁢

x

-

b

||

2

2

=

1

2

⁢

(

A

⁢

x

-

b

)

T

⁢

(

A

⁢

x

-

b

)

=

1

2

⁢

(

x

T

⁢

A

T

⁢

A

⁢

x

-

2

⁢

x

T

⁢

A

T

⁢

b

+

b

T

⁢

b

)

i.e. a differentiable function of

x

. The necessary condition for a minimum is:

∇

F

(

x

)

=

0

or

∂

⁡

F

∂

⁡

x

i

=

0

(

i

=

1

,

…

,

n

)

These equations are called the normal equations , which become in our case:

A

T

⁢

A

⁢

x

=

A

T

⁢

b

The solution

x

=

(

A

T

⁢

A

)

-

1

⁢

A

T

⁢

b

is usually computed with the following algorithm: First (the

lower triangular

portion of) the

symmetric matrix

A

T

⁢

A

is computed, then its

Cholesky decomposition

L

⁢

L

T

. Thereafter one solves

L

⁢

y

=

A

T

⁢

b

for

y

and finally

x

is computed from

L

T

⁢

x

=

y

.

Unfortunately

A

T

⁢

A

is often

ill-conditioned

and strongly influenced by roundoff errors (see [Golub89]). Other methods which do not compute

A

T

⁢

A

and solve

A

⁢

x

≈

b

directly are

QR decomposition

and

singular value decomposition

.

References

•

Originally from The Data Analysis Briefbook (

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

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

Title

normal equations

Canonical name

NormalEquations

Date of creation

2013-03-22 12:04:28

Last modified on

2013-03-22 12:04:28

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

7

Author

akrowne (2)

Entry type

Definition

Classification

msc 65-00

Classification

msc 15-00

Generated on Sat Feb 10 12:34:45 2018 by

LaTeXML
proof of existence and uniqueness of singular value decomposition

proof of existence and uniqueness of singular value decomposition

Proof of

existence

and uniqueness of SVDFernando Sanz Gamiz

Proof.

To prove existence of the SVD, we isolate the

direction

of the
largest action of

A

∈

ℂ

m

×

n

, and then proceed by

induction

on the

dimension

of

A

. We will denote

hermitian

conjugation by

T

. Norms for
vectors in

ℂ

n

will be the usual

euclidean

2-norm

∥

⋅

∥

=

∥

⋅

∥

2

and for matrix the induced by norm of vectors.

Let

σ

1

=

∥

A

∥

. By a

compactness

argument

, there must be
vectors

v

1

∈

ℂ

n

,

u

1

*

∈

ℂ

m

with

∥

v

1

∥

=

1

,

∥

u

1

*

∥

=

σ

1

and

u

1

*

=

A

⁢

v

1

.

Normalize

u

1

*

by
setting

u

1

=

u

1

*

/

∥

u

1

*

∥

and consider any

extensions

of

v

1

to an

orthonormal basis

{

v

i

}

of

ℂ

n

and of

u

1

to an
orthonormal basis

{

u

j

}

of

ℂ

m

; let

U

1

and

V

1

denote
the

unitary matrices

with columns

{

v

i

}

and

{

u

j

}

respectively. Then we have

U

1

T

⁢

A

⁢

V

1

=

S

=

(

σ

1

w

T

0

B

)

where

0

is a

column vector

of dimension

m

-

1

,

w

T

is
a

row vector

of dimension

n

-

1

, and

B

is a matrix of dimension

(

m

-

1

)

×

(

n

-

1

)

. Now,

∥

(

σ

1

w

T

0

B

)

⁢

(

σ

1

w

)

∥

≥

σ

1

2

+

w

2

=

(

σ

1

2

+

w

2

)

1

/

2

⁢

∥

(

σ

1

w

)

∥

so that

∥

S

∥

≥

(

σ

1

2

+

w

2

)

1

/

2

. But

U

1

and

V

1

are unitary matrix, hence

∥

S

∥

=

σ

1

; it
therefore implies

w

=

0

.

If

n

=

1

or

m

=

1

we are done. Otherwise the

submatrix

B

describes the action of

A

on the

subspace

orthogonal

to

v

1

. By
the

induction hypothesis

B

has an SVD

B

=

U

2

⁢

Σ

2

⁢

V

2

T

. Now it
is easily verified that

A

=

U

1

⁢

(

1

0

0

U

2

)

⁢

(

σ

1

0

0

Σ

2

)

⁢

(

1

0

0

V

2

)

T

⁢

V

1

T

is an SVD of

A

. completing the proof of existence.

For the uniqueness let

A

=

U

⁢

Σ

⁢

V

T

a SVD for

A

and let

e

i

denote the i-th,

i

=

1

⁢

⋯

⁢

m

⁢

i

⁢

n

⁢

(

m

,

n

)

vector of the canonical base of

ℂ

n

. As

U

and

V

are unitary,

∥

A

⁢

e

i

∥

=

σ

i

2

, so each

σ

i

is uniquely determined.

∎

Title

proof of existence and uniqueness of singular value decomposition

Canonical name

ProofOfExistenceAndUniquenessOfSingularValueDecomposition

Date of creation

2013-03-22 17:07:46

Last modified on

2013-03-22 17:07:46

Owner

fernsanz (8869)

Last modified by

fernsanz (8869)

Numerical id

12

Author

fernsanz (8869)

Entry type

Proof

Classification

msc 65-00

Classification

msc 15-00

Generated on Sat Feb 10 12:34:48 2018 by

LaTeXML
triangular matrix

triangular matrix

1

Triangular Matrix

Let

n

be a positive integer.

An

upper triangular matrix

is of the form:

[

a

11

a

12

a

13

⋯

a

1

⁢

n

0

a

22

a

23

⋯

a

2

⁢

n

0

0

a

33

⋯

a

3

⁢

n

⋮

⋮

⋮

⋱

⋮

0

0

0

⋯

a

n

⁢

n

]

An upper triangular matrix is sometimes also called

right triangular

.

A

lower triangular matrix

is of the form:

[

a

11

0

0

⋯

0

a

21

a

22

0

⋯

0

a

31

a

32

a

33

⋯

0

⋮

⋮

⋮

⋱

⋮

a

n

⁢

1

a

n

⁢

2

a

n

⁢

3

⋯

a

n

⁢

n

]

A lower triangular matrix is sometimes also called

left triangular

.

Note that upper triangular matrices and lower triangular matrices must be

square matrices

.

A

triangular matrix

is a matrix that is an upper triangular matrix or lower triangular matrix. Note that some matrices, such as the

identity matrix

, are both upper and lower triangular. A matrix is upper and lower triangular simultaneously if and only if it is a

diagonal matrix

.

Triangular matrices allow numerous algorithmic shortcuts in many situations. For example, if

A

is an

n

×

n

triangular matrix, the equation

A

⁢

x

=

b

can be solved for

x

in at most

n

2

operations.

In fact, triangular matrices are so useful that much computational

linear algebra

begins with factoring (or decomposing) a general matrix or matrices into triangular form. Some matrix factorization methods are the

Cholesky factorization

and the

LU-factorization

. Even including the factorization step, enough later operations are typically avoided to yield an overall time savings.

2

Properties

Triangular matrices have the following

properties

( “triangular” with either “upper” or “lower” uniformly):

•

The

inverse

of a triangular matrix is a triangular matrix.

•

The product of two triangular matrices is a triangular matrix.

•

The

determinant

of a triangular matrix is the product of the

diagonal

elements.

•

The

eigenvalues

of a triangular matrix are the diagonal elements.

The last two properties follow easily from the

cofactor expansion

of the triangular matrix.

Title

triangular matrix

Canonical

name

TriangularMatrix

Date of creation

2013-03-22 12:11:40

Last modified on

2013-03-22 12:11:40

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

13

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 65-00

Classification

msc 15-00

Defines

upper triangular

Defines

lower triangular

Defines

upper triangular matrix

Defines

lower triangular matrix

Defines

right triangular

Defines

right triangular matrix

Defines

left triangular

Defines

left triangular matrix

Generated on Sat Feb 10 12:34:50 2018 by

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