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Conversely, if M is positive definite, it has an eigendecomposition P < sup >− 1 </ sup > DP where P is unitary, D diagonal, and all diagonal elements of D are real and positive.

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## Some Related Sentences

Conversely and if

__Conversely__

**,**

__if__statistics were uncovered which contradicted a cherished theory

**,**the sources were denounced as faulty

**.**

__Conversely__

**,**experimenters would consider as impressive such statements as the following

**,**which

**,**

__if__they turned out to be hits

**,**

**are**so unusual as to be really significant:

__Conversely__

**,**

__if__he gives a heavy rating to his own reading

**,**

**and**finds more accurate facts in

**it**than in the others

**,**a point

**is**chalked up for the intrinsic

**,**objective meaningfulness

**of**this type

**of**mediumistic material

**.**

__Conversely__

**,**a subset R defines a binary function

__if__

**and**only

__if__

**,**for any x in X

**and**y in Y

**,**there exists a unique z in Z such that ( x

**,**y

**,**z ) belongs to R

**.**

__Conversely__

**,**

__if__a Boolean ring A

**is**given

**,**we can turn

**it**into a Boolean algebra by defining x ∨ y := x + y + ( x · y )

**and**x ∧ y := x · y

**.**

__Conversely__

**,**

__if__the second theory

**is**correct

**,**then the Athenians were merely reacting to the Persians attacking them

**.**

__Conversely__the equilibrium position

**is**said to be " far to the left "

__if__hardly any product

**is**formed from the reactants

**.**

__Conversely__

**,**

__if__ƒ: C → C

**is**a function which

**is**differentiable when regarded as a function on R

**<**

**sup**

**>**2

**</**

**sup**>, then ƒ

**is**complex differentiable

__if__

**and**only

__if__the Cauchy – Riemann equations hold

**.**

__Conversely__

**,**

__if__the amount

**of**fat

**is**increased

**,**increasing the number

**of**calories ingested will only add weight

**.**

__Conversely__

**,**

__if__the source

**of**waves

**is**moving away from the observer

**,**each wave

**is**emitted from a position farther from the observer than the previous wave

**,**so the arrival time between successive waves

**is**increased

**,**reducing the frequency

**.**

__Conversely__every partition

**of**comes from

**an**equivalence relation in this way

**,**according to which

__if__

**and**only

__if__

**and**belong to the same set

**of**the partition

**.**

If neither A nor B includes the idea

**of**existence**,**then " some A**are**B " simply adjoins A to B**.**__Conversely__**,**__if__A or B do include the idea**of**existence in the way that " triangle " contains the idea " three angles equal to two right angles ", then " A exists "**is**automatically true**,****and**we have**an**ontological proof**of**A's existence**.**__Conversely__

**,**

__if__the measured potential

**is**negative

**,**the electrode

**is**more oxidizing than the SHE ( such as Zn in ZnSO

**<**sub

**>**4

**</**sub

**>**

**where**the standard electrode potential

**is**− 0

**.**76 V ).

__Conversely__

**,**

__if__the firm

**is**able to get bulk discounts

**of**

**an**input

**,**then

**it**could have economies

**of**scale in some range

**of**output levels even

__if__

**it**

**has**decreasing returns in production in that output range

**.**

__Conversely__

**,**daylight stock shot in daylight will look normal

**,**but orange

__if__shot under tungsten lights

**.**

__Conversely__

**,**

__if__the data

**are**sparse — that

**is**

**,**

__if__only K out

**of**N Fourier coefficients

**are**nonzero — then the complexity can be reduced to O ( K log N log ( N / K )),

**and**this

**has**been demonstrated to lead to practical speedups compared to

**an**ordinary FFT for N / K

**>**32 in a large-N example ( N = 2

**<**

**sup**

**>**22

**</**

**sup**>) using a probabilistic approximate algorithm ( which estimates the largest K coefficients to several decimal places ).

Conversely and M

__Conversely__

**,**any Hermitian

**positive**semidefinite matrix

__M__can be written as

__M__= A

**<**

**sup**>*</

**sup**

**>**A ; this

**is**the Cholesky decomposition

**.**

__Conversely__

**,**

**if**∇

**is**a connection on E

**and**A

**is**a one-form on

__M__with values in End ( E ), then ∇+ A

**is**a connection on E

**.**

__Conversely__

**,**one can show that given a manifold

__M__

**and**a principal H-bundle

**P**over

__M__

**,**

**and**a 1-form η with these properties

**,**then

**P**

**is**locally isomorphic as

**an**H-bundle to the principal homogeneous bundle G → G / H

**.**

A section Γ

**of**dπ: TP / G → TM such that Γ: TM → TP / G**is**a linear morphism**of**vector bundles over__M__**,**can be identified with a principal connection in**P****.**__Conversely__**,**a principal connection as defined above gives rise to such a section Γ**of**TP / G**.**__Conversely__

**,**given a left kG module

__M__

**,**then

__M__

**is**a k vector space

**,**

**and**multiplication with

**an**element g

**of**G yields a k-linear automorphism

**of**

__M__( since g

**is**invertible in kG ), which describes a group homomorphism G → GL (

__M__).

__Conversely__

**,**he refused to share information in his APL-provided files when Ohio Governor James

__M__

**.**Cox requested

**it**

**.**

Suppose S

**is****an**immersed submanifold**of**__M__**.**If the inclusion map i: S →__M__**is**closed then S**is**actually**an**embedded submanifold**of**__M__**.**__Conversely__**,****if**S**is****an**embedded submanifold which**is**also a closed subset then the inclusion map**is**closed**.**__Conversely__

**,**Austereo's Triple

__M__group

**of**stations have suffered heavily from increased competition from new station Vega 95

**.**3

**and**Vega 91

**.**5 in Sydney

**and**Melbourne respectively

**,**with the stations in Melbourne

**and**Sydney recording some

**of**their lowest ratings

**of**

**all**time

**.**

__Conversely__

**,**

**if**the holonomy group

**of**the Riemannian manifold

__M__

**is**contained in Sp ( k ), choose complex structures I

**<**sub

**>**x

**</**sub >, J

**<**sub

**>**x

**</**sub

**>**

**and**K

**<**sub

**>**x

**</**sub

**>**on T

**<**sub

**>**x

**</**sub

**>**

__M__which make T

**<**sub

**>**x

**</**sub

**>**

__M__into a quaternionic vector space

**.**

:

__Conversely__**,**suppose that φ**is**a smooth real-valued function on__M__in a neighborhood**of**p**.**Since every smooth function**has**a local coordinate expression**,**we may express φ as a function in the coordinates**.**
To summarize

**,**__M__**is**a symmetric space G / K with a compact isotropy group K**.**__Conversely__**,**symmetric spaces with compact isotropy group**are**Riemannian symmetric spaces**,**although not necessarily in a unique way**.**
Statement 4 then implies that a basis

**of**__M__/ mM lifts to a minimal set**of**generators**of**__M__**.**__Conversely__**,**every minimal set**of**generators**of**__M__**is**obtained in this way**,****and**any two such sets**of**generators**are**related by**an**invertible matrix with entries in the ring**.**

Conversely and is

__Conversely__

**,**

**an**adiabatic process that

__is__irreversible

**and**extracts no work

__is__in

**an**isenthalpic process

**,**such as viscous drag

**,**progressing towards a nonnegative change in entropy

**.**

__Conversely__

**,**British English favours fitted as the past tense

**of**fit generally

**,**whereas the preference

**of**American English

__is__more complex: AmEng prefers fitted for the metaphorical sense

**of**having made

**an**object " fit " ( i

**.**e., suited ) for a purpose ; in spatial transitive contexts

**,**AmEng uses fitted for the sense

**of**having made

**an**object conform to

**an**unchanged object that

**it**surrounds ( e

**.**g., " fitted X around Y ") but fit for the sense

**of**having made

**an**object conform to

**an**unchanged object that surrounds

**it**( e

**.**g., " fit X into Y ");

**and**for the spatial senses ( both intransitive

**and**transitive )

**of**having been matching with respect to contour

**,**with no alteration

**of**either object implied

**,**AmEng prefers fit (" The clothes fit

**.**

__Conversely__the use

**of**true brass seems to have declined in Western Europe during this period in favour

**of**gunmetals

**and**other mixed alloys but by the end

**of**the first Millennium AD brass artefacts

**are**found in Scandinavian graves in Scotland

**,**brass was being used in the manufacture

**of**coins in Northumbria

**and**there

__is__archaeological

**and**historical evidence for the production

**of**brass in Germany

**and**The Low Countries areas rich in calamine ore which would remain important centres

**of**brass making throughout the medieval period

**,**especially Dinant – brass objects

**are**still collectively known as dinanterie in French

**.**

__Conversely__

**,**because space

__is__expanding

**,**

**and**more distant objects

**are**receding ever more quickly

**,**light emitted by us today may never " catch up " to very distant objects

**.**

__Conversely__

**,**modern biological sciences ( including even concepts such as molecular ecology )

**are**intimately entwined

**and**dependent on the methods developed through biotechnology

**and**what

__is__commonly thought

**of**as the life sciences industry

**.**

__Conversely__

**,**the islands do not receive the upwellings ( cold streams ) that affect the West African coast

**,**so the air temperature

__is__cooler than in Senegal

**,**but the sea

__is__actually warmer

**,**because the orographic relief

**of**some islands

**,**as Sao Miguel with steep mountains

**,**cover

**it**with rich woods

**and**luxuriant vegetation

**where**the dense moisture condense

**and**soak the plants

**,**rocks

**,**soil

**,**logs

**,**moss etc

**.**

__Conversely__in francophone Canada

**,**one hundredth

**of**a Canadian dollar

__is__informally called a sou ( penny ), though cent

__is__official in both English

**and**French

**.**

__Conversely__

**,**

**an**absence

**of**secure cycle-parking

__is__a recurring complaint by cyclists from cities with low modal share

**of**cycling

**.**

* The spectrum

**of**any bounded linear operator on a Banach space__is__a nonempty compact subset**of**the complex numbers C**.**__Conversely__**,**any compact subset**of**C arises in this manner**,**as the spectrum**of**some bounded linear operator**.**0.123 seconds.