[permalink] [id link]

If the constraints of a linear algebra problem do not allow a general matrix to be conveniently reduced to a triangular one, reduction to Hessenberg form is often the next best thing.

from
Wikipedia

## Some Related Sentences

If and constraints

__If__

**the**media agency responsible for

**the**authorized production allows material from fans

**,**what

**is**

**the**limit before legal

__constraints__from actors

**,**music

**,**and other considerations

**,**come into play?

__If__

**the**

**matrix**

**is**positive semidefinite

**matrix**

**,**then

**is**

**a**convex function: In this case

**the**quadratic program has

**a**global minimizer if there exists some feasible vector ( satisfying

**the**

__constraints__) and if

**is**bounded below on

**the**feasible region

**.**

__If__

**the**

__constraints__don't couple

**the**variables too tightly

**,**

**a**relatively simple attack

**is**

**to**change

**the**variables so that

__constraints__are unconditionally satisfied

**.**

__If__

**the**person being questioned wouldn't necessarily consent

**to**those

__constraints__

**,**

**the**question

**is**fallacious

**.**

__If__there had been other

__constraints__

**,**triggers or cascades

**,**every single change operation would have been checked in

**the**same way as

**the**above before

**the**transaction was committed

**.**

__If__grammars differ only by having different rankings

**of**CON

**,**then

**the**set

**of**possible human languages

**is**determined by

**the**

__constraints__that exist

**.**

__If__

**the**dust disk

**is**instead being generated from

**the**outer debris disk

**,**rather than from collisions in an asteroid belt

**,**then no

__constraints__on

**the**planet's orbital eccentricity are needed

**to**explain

**the**dust distribution

**.**

__If__we take basis vectors for V

**,**those become non-commuting variables ( or indeterminants ) in T ( V ), subject

**to**no

__constraints__( beyond associativity

**,**

**the**distributive law and K-linearity ).

__If__there are

**,**say

**,**m

__constraints__then

**the**zero in

**the**north-west corner

**is**an m × m block

**of**zeroes

**,**and there are m border rows at

**the**top and m border columns at

**the**left

**.**

__If__vulcanoids are found

**not**

**to**exist

**,**this would place different

__constraints__on planet formation and suggest that other processes have been at work in

**the**inner Solar System

**,**such as planetary migration clearing out

**the**area

**.**

__If__there are n

__constraints__holding n different expectation values constant

**,**then

**the**manifold

**is**n-dimensional

**.**

__If__

**the**labels are all congruent rectangles

**,**

**the**corresponding 2-SAT instance can

**be**shown

**to**have only linearly many

__constraints__

**,**leading

**to**near-linear time algorithms for finding

**a**labeling

**.**

__If__there

**is**

**a**unifying theme that runs through most

**of**managerial economics

**,**it

**is**

**the**attempt

**to**optimize business decisions given

**the**firm's objectives and given

__constraints__imposed by scarcity

**,**for example through

**the**use

**of**operations research

**,**mathematical programming

**,**game theory for strategic decisions

**,**and other computational methods

**.**

__If__we have more than n + 1

__constraints__( n being

**the**degree

**of**

**the**polynomial ), we can still run

**the**polynomial curve through those

__constraints__

**.**

__If__humans escape these

__constraints__

**,**it

**is**because in our case

**,**listeners are primarily interested in mental states

**.**

__If__

**the**objective function

**is**

**a**ratio

**of**

**a**concave and

**a**convex function ( in

**the**maximization case ) and

**the**

__constraints__are convex

**,**then

**the**

**problem**can

**be**transformed

**to**

**a**convex optimization

**problem**using fractional programming techniques

**.**

__If__

**the**person being questioned wouldn't necessarily consent

**to**those

__constraints__

**,**

**the**question

**is**fallacious

**.**

__If__

**a**syntax error occurs or if any

__constraints__are violated

**,**

**the**new row

**is**

**not**added

**to**

**the**table and an error returned instead

**.**

__If__it

**is**

**not**possible

**to**build

**a**second antenna for

**the**second transmitter due

**to**space

__constraints__

**,**then

**the**diplexer

**is**used permanently

**.**

If and linear

__If__

**the**Af bond

**is**

__linear__then there are three reasonable positions for

**the**hydrogen atoms: ( 1 ) The hydrogen atoms are centered and hence all lie on

**a**sheet midway between

**the**oxygen sheets ; ;

__If__Af are

**the**projections associated with

**the**primary decomposition

**of**T

**,**then each Af

**is**

**a**polynomial in T

**,**and accordingly if

**a**

__linear__operator U commutes with T then U commutes with each

**of**

**the**Af

**,**i.e.

**,**each subspace Af

**is**invariant under U

**.**

__If__T

**is**

**a**

__linear__operator on an arbitrary vector space and if there

**is**

**a**monic polynomial P such that Af

**,**then parts ( A ) and ( B )

**of**Theorem 12 are valid for T with

**the**proof which we gave

**.**

__If__X

**is**

**a**Banach space and K

**is**

**the**underlying field ( either

**the**real or

**the**complex numbers ), then K

**is**itself

**a**Banach space ( using

**the**absolute value as norm ) and we can define

**the**continuous dual space as X ′ = B ( X

**,**K ),

**the**space

**of**continuous

__linear__maps into K

**.**

The tensor product X ⊗ Y from X and Y

**is****a**K-vector space Z with**a**bilinear function T: X × Y → Z which has**the**following universal property:__If__T ′: X × Y → Z ′**is**any bilinear function into**a**K-vector space Z ′, then only**one**__linear__function f: Z → Z ′ with exists**.**
*

__If__**is****the**norm ( usually noted as ) defined in**the**sequence space ℓ < sup >∞</ sup >**of**all bounded sequences ( which also matches**the**non-linear distance measured as**the**maximum**of**distances measured on projections into**the**base subspaces**,**without requiring**the**space**to****be**isotropic or even just__linear__**,**but only continuous**,**such norm being definable on all Banach spaces ), and**is**lower**triangular**non-singular ( i**.**e., ) then__If__

**the**resulting

__linear__differential equations have constant coefficients

**one**can take their Laplace transform

**to**obtain

**a**transfer function

**.**

__If__we assume

**the**controller C

**,**

**the**plant P

**,**and

**the**sensor F are

__linear__and time-invariant ( i

**.**e., elements

**of**their transfer function C ( s ), P ( s ), and F ( s )

**do**

**not**depend on time ),

**the**systems above can

**be**analysed using

**the**Laplace transform on

**the**variables

**.**

__If__

**the**function f

**is**

**not**

__linear__( i

**.**e

**.**its graph

**is**

**not**

**a**straight line ), however

**,**then

**the**change in y divided by

**the**change in x varies: differentiation

**is**

**a**method

**to**find an exact value for this rate

**of**change at any given value

**of**x

**.**

__If__

**the**

**matrix**entries are real numbers

**,**

**the**

**matrix**can

**be**used

**to**represent two

__linear__mappings:

**one**that maps

**the**standard basis vectors

**to**

**the**rows

**of**

**,**and

**one**that maps them

**to**

**the**columns

**of**

**.**

__If__λ < sub > 1 </ sub >, ..., λ < sub > ν </ sub > are

**the**eigenvalues

**of**J they will

**be**resonant if

**one**eigenvalue

**is**an integer

__linear__combination

**of**two or more

**of**

**the**others

**.**

__If__I told you my son's age

**,**then there would no longer

**be**two unknowns ( variables ), and

**the**

**problem**becomes

**a**

__linear__equation with just

**one**variable

**,**that can

**be**solved as described above

**.**

Likewise

**,****a**functor from G**to****the**category**of**vector spaces**,**Vect < sub > K </ sub >,**is****a**__linear__representation**of**G**.**In**general****,****a**functor G → C can**be**considered as an " action "**of**G on an object in**the**category C**.**__If__C**is****a**group**,**then this action**is****a**group homomorphism**.**
Tensor products:

__If__C denotes**the**category**of**vector spaces over**a**fixed field**,**with__linear__maps as morphisms**,**then**the**tensor product defines**a**functor C × C → C which**is**covariant in both arguments**.**
*

__If__V**is****a**normed vector space with__linear__subspace U (**not**necessarily closed ) and if**is**continuous and__linear__**,**then there exists an extension**of**φ which**is**also continuous and__linear__and which has**the**same norm as φ ( see Banach space for**a**discussion**of****the**norm**of****a**__linear__map ).
*

__If__V**is****a**normed vector space with__linear__subspace U (**not**necessarily closed ) and if z**is**an element**of**V**not**in**the**closure**of**U**,**then there exists**a**continuous__linear__map with ψ ( x ) = 0 for all x in U**,**ψ ( z ) = 1**,**and || ψ || = 1 / dist ( z**,**U ).__If__

**the**table size n

**is**large enough

**,**

__linear__search will

**be**faster than binary search

**,**whose cost

**is**O ( log n ).

__If__

**the**list

**is**stored as an ordered array

**,**then binary search

**is**almost always more efficient than

__linear__search as with n > 8

**,**say

**,**unless there

**is**some reason

**to**suppose that most searches will

**be**for

**the**small elements near

**the**start

**of**

**the**sorted list

**.**

0.170 seconds.