otheruses2 Identity In mathematics , the term identity has several different important meanings An identity is a relation which is Tautology logic tautologically true. This means that whatever the number or value may be, the answer stays the same. For example, algebraically, this occurs if an equation is satisfied for all values of the involved variables. Definitions are often indicated by the triple bar symbol , such as A sup 2 sup x x . The symbol can also be used with other meanings, but these can usually be interpreted in some way as a definition, or something which is otherwise tautologically true for example, a congruence relation . In algebra , an identity or identity element of a set S with a binary operation is an element e that, when combined with any element x of S , produces that same x . That is, nowrap 1 e x x e x for all x in S . An example of this is the identity matrix . The identity ... for instance, the identity permutation is the identity element in the Group mathematics group of permutations ... simple Identitymathematics sv Identitet matematik ta uk ... x in S . This function serves as the identity element in the set of all functions from S to itself with respect to function composition . Examples Identity relation A common example of the first meaning is the trigonometric identity math sin 2 theta cos 2 theta equiv 1 , math which is true for all ... false when math theta 2 , math . See also list of mathematical identities . Identity element The concepts of additive identity and multiplicative identity are central to the Peano axioms . The number 0 is the additive identity for integers, real numbers, and complex numbers. For the real numbers, for all ..., The number 1 is the multiplicative identity for integers, real numbers, and complex numbers. For the real ... 1 times 1 1. , math Identity function A common example of an identity function is the identity ... EquationSolver A webpage that can test a suggested identity and return a true false verdict . Category ... more details
identity , belief in membership of a nation MathematicsIdentity element , a special element in a set or structure with respect to an operator Identity function , a function that does not alter its argument Identitymathematics , an equality that holds regardless of the values of its variables Identity matrix , a square matrix with ones on the main diagonal , zeros elsewhere Business Accounting identity , calculation or measurement that must be true regardless of the value of its variables Corporate identity , persona of a corporation by way of branding or use of trademarks Computer science Digital identity , representation of a set of claims made by one digital subject about itself or another ...wiktionary tocright Identity may refer to Philosophical topics Identity philosophy , also called sameness, is whatever makes an entity definable and recognizable Law of identity , principle of logic stating that an object is the same as itself Personal identity philosophy , refers to the numerical identity continuity of existence of persons through time Specifications of persons Identity change Identity document Identity theft , the deliberate appropriation of someone else s identity without that person ... Gender identity , also known as core gender identity, the gender s or lack thereof, that a person self identifies Identity formation , the process of the development of the distinct personality of an individual Identity social science , umbrella term used to describe individuality, personal identity, social identity, and cultural identity in psychology, sociology, and philosophy Persona , a social role or a character played by an actor with oneself Sexual orientation identity , describes how persons identify their own sexuality Group expression and affiliation Christian Identity , a Christian religious movement Cultural identity , person s self affiliation or categorization by others as a member of a cultural group Identity politics , refers to political arguments that focus ... more details
uses see Mathematics disambiguation and Math disambiguation . File Euclid.jpg thumb Euclid , Greek ... . ref Mathematics from Greek language Greek m th ma knowledge, study, learning is the study ..., then mathematical reasoning often provides insight or predictions. Through the use of abstraction mathematics abstraction and logic al reasoning , mathematics developed from counting , calculation , measurement .... Practical mathematics has been a human activity for as far back as History of Mathematics written records exist. Logic Rigorous arguments first appeared in Greek mathematics , most notably in Euclid Euclid s Euclid s Elements Elements . Mathematics developed at a relatively slow pace until the Renaissance ... History of Mathematics 1. Newton and Leibniz , BBC Radio 4 , 27 09 2010. ref Carl Friedrich Gauss 1777 1855 referred to mathematics as the Queen of the Sciences . ref Waltershausen ref Benjamin Peirce 1809 1880 called mathematics the science that draws necessary conclusions . ref Peirce, p. 97. ref David Hilbert said of mathematics We are not speaking here of arbitrariness in any sense. Mathematics ..., Birkh user 1992 . ref Albert Einstein 1879 1955 stated that as far as the laws of mathematics ... . ref name certain Mathematics is used throughout the world as an essential tool in many fields, including natural science , engineering , medicine , and the social sciences . Applied mathematics , the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires ... mathematics , or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered. ref Peterson ref Etymology The word mathematics comes from the ancient ... mean to learn . The word mathematics in Greek came to have the narrower and more technical meaning ... until around 1700, the term mathematics more commonly meant astrology or sometimes astronomy ... more details
Dual identity can refer to A secret identity , such as Clark Kent and Superman In mathematics, the coidentity of a dual group object or the counit of a coalgebra dab ... more details
In mathematics , an approximation to the identity refers to a sequence or net that converges to the identity in some algebra. Specifically, it can mean Nascent delta function , most commonly Mollifier , more narrowly Approximate identity , more abstractly disambig ... more details
In mathematics , the term Fibonacci s identity may refer either to the Brahmagupta Fibonacci identity in algebra, showing that the set of all sums of two squares is closed under multiplication or to the Cassini and Catalan identities on Fibonacci number s. Category Mathematical disambiguation Category mathematical identities disambig ... more details
In mathematics the additive identity of a Set mathematics set which is equipped with the operation mathematics operation of addition is an element mathematics element which, when added to any element x ... mathematics , but additive identities occur in other mathematical structures where addition is defined, such as in group mathematics groups and ring mathematics rings . Elementary examples The additive identity familiar from elementary mathematics is zero, denoted 0 number 0 . For example ... identity 1 are equal, or 0 1. Let r be any element mathematics element of R . Then r r 1 r 0 0, proving ... s Q , the real number s R , or the complex number s C , the additive identity is 0. Thus for any one of these number s n , n 0 n 0 n . The additive identity is zero in an addition problem Formal definition Let N be a Set mathematics set which is closed under the operation mathematics operation of addition , denoted . An additive identity for N is any element e such that for any element n in N , e n n n e . Further examples In a group mathematics group the additive identity is the identity element of the group, is often denoted 0, and is unique see below for proof . A ring mathematics ring or field mathematics field is a group under the operation of addition and thus these also have a unique additive identity 0. This is defined to be different from the multiplicative identity 1 number 1 if the ring or field has more than one element. If the additive identity and the multiplicative identity are the same, then the ring is trivial mathematics trivial proved below . In the ring M sub m n sub R of m by n matrix mathematics matrices over a ring R , the additive identity is denoted 0 and is the m by n matrix whose entries consist entirely of the identity element 0 in R . For example, in the 2 by 2 matrices over the integers M sub 2 sub Z the additive identity is math 0 begin pmatrix 0 & 0 0 & 0 end pmatrix . math In the quaternions , 0 is the additive identity. In the ring of function ... more details
In mathematics , an identity element or neutral element is a special type of element of a Set mathematics ... 0 under most definitions of GCD R sup n sup multiplication 1 number 1 m by n matrix mathematics matrices addition zero matrix matrix of all zeroes n by n square matrix mathematics matrices multiplication I sub n sub identity matrix matrix with 1 on diagonal br and 0 elsewhere all function mathematics functions from a set M to itself function composition identity function all function mathematics ... with them. This is used for group mathematics group s and magma algebra related concepts . The term identity element is often shortened to identity as will be done in this article when there is no possibility ... magma . Then an element e of S is called a left identity if e a a for all a in S , and a right identity if a e a for all a in S . If e is both a left identity and a right identity, then it is called a two sided identity , or simply an identity . An identity with respect to addition is called an additive identity often denoted as 0 and an identity with respect to multiplication is called a multiplicative identity often denoted as 1 . The distinction is used most often for sets that support both binary operations, such as ring mathematics ring s. The multiplicative identity is often called the unit in the latter context, where, unfortunately, a unit ... class wikitable style margin 1em auto 1em auto set operation identity real number s addition 0 number 0 real number s multiplication 1 number 1 real number s a sup b sup exponentiation 1 right identity ... subsets of a Set mathematics set M intersection M sets union empty set boolean logic logical and truth ... identity br and no two sided identity Properties As the last example shows, it is possible for S , to have several left identities. In fact, every element can be a left identity. Similarly, there can be several right identities. But if there is both a right identity and a left identity ... more details
About the Banach algebra concept Approximation to the identity disambiguation Approximation to the identity In functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring possibly without an identity that acts as a substitute for an identity element. More precisely, a right approximate identity in a Banach algebra , A , is a net mathematics net or a sequence math ,e lambda lambda in Lambda , math such that for every element, a , of A , the net or sequence math ,ae lambda lambda in Lambda , math has limit a . Similarly, a left approximate identity is a net math ,e lambda lambda in Lambda , math such that for every element, a , of A , the net or sequence math ,e lambda a lambda in Lambda , math has limit a . An approximate identity is a right approximate identity which is also a left approximate identity. For C algebra s, a right or left approximate identity is the same as an approximate identity. Every C algebra has an approximate identity of positive element positive elements of norm &le 1 indeed, the net of all positive elements of norm &le 1 in A with its natural order always suffices. This is called the canonical approximate identity of a C algebra. Approximate identities of C algebras are not unique. For example, for compact operators acting on a Hilbert space, the net consisting of finite rank projections would be another approximate identity. An approximate identity in a convolution algebra plays the same role as a sequence of function approximations to the Dirac delta function which is the identity element for convolution . For example the Fej r kernel s of Fourier series theory give rise to an approximate identity. Ring theory In ring theory an approximate identity is defined in a similar way, except that the ring is given the discrete topology so that a ae sub sub for some . A module over a ring with approximate identity is called non degenerate if for every m in the module there is some with m me sub sub . See also ... more details
distinguish Null function Empty function Unreferenced date December 2009 In mathematics , an identity function , also called identity map or identity transformation , is a function mathematics function that always returns the same value that was used as its argument. In terms of equation s, the function is given by f x x . Definition Formally, if M is a Set mathematics set , the identity function f on M is defined to be that function with domain mathematics domain and codomain M which satisfies f x x for all elements x in M . In other words, the function assigns to each element x of M the element x of M . The identity function f on M is often denoted by id sub M sub . In terms of set theory , where a function is defined as a particular kind of binary relation , the identity function is given by the identity relation , or diagonal of M . Algebraic property If f M N is any function, then we have f small o small id sub M sub f id sub N sub small o small f where small o small denotes function composition . In particular, id sub M sub is the identity element of the monoid of all functions from M to M . Since the identity element of a monoid is unique , one can alternately define the identity function on M to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory , where the endomorphism s of M need not be functions. Properties The identity function is a linear map linear operator , when applied to vector space s. The identity function on the positive integer s is a completely multiplicative ... vector space the identity function is represented by the identity matrix I sub n sub , regardless of the Basis linear algebra basis . In a metric space the identity is trivially an isometry . An object ... type C sub 1 sub . See also Inclusion map DEFAULTSORT Identity Function Category Functions and mappings Category Elementary mathematics Category Basic concepts in set theory Category Types of functions ... more details
Unreferenced date December 2009 In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. Unlike ... Jacobi identity. It is named after the Germany German mathematician Carl Gustav Jakob Jacobi . Definition A binary operation math times math on a Set mathematics set math S math possessing a commutative binary operation math math with additive identity 0 satisfies the Jacobi identity if math a times ... algebra , the objects that obey the Jacobi identity are infinitesimal motions. When acting on an operator ... Identity math A , B , C A , B , C B , A , C , math can then be translated into words the infinitesimal ... . Examples The Jacobi identity is satisfied by the multiplication bracket operation on Lie ... the Jacobi identity in common use. Because of this the Jacobi identity is often expressed using ... antisymmetric , the Jacobi identity admits two equivalent reformulations. Defining the adjoint representation ..., the identity becomes math operatorname ad x y,z operatorname ad xy,z y, operatorname ad xz . math Thus, the Jacobi identity for Lie algebras simply becomes the assertion that the action of any element on the algebra is a derivation abstract algebra derivation . This form of the Jacobi identity ... identity is equivalent to the following identity between the operators of the Adjoint representation ... ad y . math This identity implies that the map sending each element to its adjoint action is a Lie algebra homomorphism of the original algebra into the Lie algebra of its derivations. A similar identity, called the Commutators Identities Hall Witt identity , exists for the commutator s in group mathematics groups . In analytical mechanics , Jacobi identity is satisfied by Poisson bracket s, while in quantum mechanics it is satisfied by operator commutators. See also Super Jacobi identity Hall Witt identity DEFAULTSORT Jacobi Identity Category Lie algebras Category Mathematical identities Category ... more details
In mathematics , a trace identity is any equation involving the trace linear algebra trace of a matrix mathematics matrix . For example, the Cayley Hamilton theorem says that every matrix satisfies its own characteristic polynomial . Trace identities are invariant under simultaneous Conjugation mathematics conjugation . They are frequently used in the invariant theory of n n matrices to find the generators and Relation mathematics relations of the invariant theory ring of invariants , and therefore are useful in answering questions similar to that posed by Hilbert s fourteenth problem . Examples By the Cayley Hamilton theorem , all matrices satisfy math rm tr A n rm tr A rm tr A n 1 cdots 1 n det A 0. , math All matrices satisfy math rm tr A rm tr A T . , math See also Invariant polynomial Category linear algebra Category invariant theory Linear algebra stub ... more details
mathematics equality . More mundanely, an identity in mathematics may be an equation that holds ... is expressed in mathematics with the symbol, e.g., a b , or Clark Kent Superman. See also Identity ...In philosophy , identity , from Lang la identitas sameness , is the relation each thing bears just to itself. ref Stanford Encyclopedia of Philosophy http plato.stanford.edu entries identityIdentity , First ... of Philosophy, 2nd Edition, CUP 1995 ref The notion of identity gives rise to many philosophical problems, including the identity of indiscernibles if x and y share all their properties, are they one and the same thing? , and questions about identity and change change and Personal identity philosophy personal identity over time what has to be the case for a person x at one time and a person y ... of identity from the more well known notion of identity in use in psychology and the social science ... other i.e. just in case x y . The identity social science sociological notion of identity , by contrast ... of a person that make them unique, or qualitatively different from others e.g. cultural identity , gender identity , national Identity , online identity and processes of identity formation . Logic of identity In logic , the Binary relation Sets versus classes identity relation also called equality is normally defined as the binary relation that holds only between a thing and itself. That is, identity ... if x is the same thing as y . Identity is transitive relation transitive , symmetric relation ... x and y , if x y then necessarily y x . That is, identity does not hold ... 2 because 1 1 and 2 are different guises for the same number. Similarly, for personal identity over time, me today and me yesterday are different guises for the same person. Metaphysics of identity This section ... it mean for an object to be the same, if it Identity and change change s over time? Is apple SUB t SUB ... of Theseus example, in what way is it the same? The Law of identity originates from classical antiquity ... more details
In mathematics, Vaughan s identity is an identity found by harvs txt yes authorlink Robert Charles Vaughan mathematician last Vaughan first R. C. year 1977 that can be used to simplify I.M. Vinogradov Vinogradov s work on trigonometric sum s. It can be used to estimate sums of the form math sum f n Lambda n math where f is some function of positive integers n , whose values in applications are often roots of unity, and is the von Mangoldt function . Vaughan s identity has been used to simplify the proof of the Bombieri Vinogradov theorem and to study Kummer sum s. Vaughan s identity was generalized by harvtxt Heath Brown 1982 . References springer id V v130030 first S.W. last Graham citation authorlink Heath Brown last Heath Brown first D. R. title Prime numbers in short intervals and a generalized Vaughan identity journal Canad. J. Math. volume 34 year 1982 issue 6 pages 1365 1377 mr 0678676 doi 10.4153 CJM 1982 095 9 citation first R.C. last Vaughan title Sommes trigonom triques sur les nombres premiers journal C.R. Acad. Sci. Paris S r. A volume 285 year 1977 pages 981 983 mr 0498434 Category analytic number theory numtheory stub ... more details
In mathematics , Proizvolov s identity is an identity concerning sums of differences of positive integer s. The identity was posed by Vyacheslav Proizvolov as a problem in the 1985 All Union Soviet Student Olympiads harv Savchev Andreescu 2002 p 66 . To state the identity, take the first 2 N positive integers, 1, 2, 3, ..., 2 N &minus 1, 2 N , and partition them into two subsets of N numbers each. Arrange one subset in increasing order math A 1 A 2 cdots A N. math Arrange the other subset in decreasing order math B 1 B 2 cdots B N. math Then the sum math A 1 B 1 A 2 B 2 cdots A N B N math is always equal to N sup 2 sup . Example Take for example N 3. The set of numbers is then 1, 2, 3, 4, 5, 6 . Select three numbers of this set, say 2, 3 and 5. Then the sequences A and B are A sub 1 sub 2, A sub 2 sub 3, and A sub 3 sub 5 B sub 1 sub 6, B sub 2 sub 4, and B sub 3 sub 1. The sum is math A 1 B 1 A 2 B 2 A 3 B 3 2 6 3 4 5 1 4 1 4 9, math which indeed equals 3 sup 2 sup . References Citation last Savchev first Svetoslav last2 Andreescu first2 Titu year 2002 title Mathematical miniatures volume 43 series Anneli Lax New Mathematical Library publisher Mathematical Association of America isbn 088385645X . External links http www.cut the knot.org Curriculum Games ProizvolovGame.shtml Proizvolov s identity at cut the knot.org Category Recreational mathematics Category Theorems in number theory es Identidad de Proizvolov ko ... more details
merge Bochner s formula discuss Talk Bochner s formula Merger proposal date February 2009 In mathematics &mdash specifically, differential geometry &mdash the Bochner identity is an Identitymathematicsidentity concerning harmonic map s between Riemannian manifold s. The identity is named after the United States American mathematician Salomon Bochner . Statement of the result Let M and N be Riemannian manifolds and let u M N be a harmonic map. Let d denote the exterior derivative , the gradient , the Laplace Beltrami operator , Riem sub N sub the Riemann curvature tensor on N and Ric sub M sub the Ricci curvature tensor on M . Then math Delta big nabla u 2 big big nabla mathrm d u big 2 big langle mathrm Ric M nabla u, nabla u big rangle big langle mathrm Riem N u nabla u, nabla u nabla u, nabla u big rangle. math References cite journal last Eells first J coauthors Lemaire, L. title A report on harmonic maps journal Bull. London Math. Soc. volume 10 year 1978 issue 1 pages 1&ndash 68 doi 10.1112 blms 10.1.1 mr 495450 External links MathWorld urlname BochnerIdentity title Bochner identity See also Bochner s formula Category Differential geometry Category Mathematical identities ... more details
In mathematics , the identity component of a topological group G is the connected component topology connected component G sub 0 sub of G that contains the identity element of the group. Similarly, the identity path component of a topological group G is the path component of G that contains the identity element of the group. Properties The identity component G sub 0 sub of a topological group G is a closed set closed , normal subgroup of G . It is closed since components are always closed. It is a subgroup since multiplication and inversion in a topological group are continuous map topology continuous map s by definition. Moreover, for any continuous automorphism a of G we have a G sub 0 sub G sub 0 sub . It follows that G sub 0 sub is normal in G . The identity component G sub 0 sub of a topological group G need not be open set open in G . In fact, we may have G sub 0 sub e , in which case G is totally disconnected group totally disconnected . However, the identity component of a locally path connected space for instance a Lie group is always open, since it contains a path connected neighbourhood of e and therefore is a clopen set . The identity path component may in general be smaller than the identity component since path connectedness is a stronger condition than connectedness , but these agree if G is locally path connected. Component group The quotient group G G sub 0 sub is called the group of components or component group of G . Its elements are just the connected components of G . The component group G G sub 0 sub is a discrete group if and only if G sub 0 sub is open. If G is an affine algebraic group then G G sub 0 sub is actually a finite group . One may similarly define the path component group as the group of path components quotient of G by the identity path component , and in general the component group is a quotient of the path component group, but if G is locally path connected these groups agree. The path component group can also be characterized as the zeroth ... more details
analysis analytical mathematics , Euler s Identity , named for the Swiss German mathematician ... named Euler s Identity as the most beautiful theorem in mathematics . ref Nahin, 2006, http books.google.com ... 1 e 2 pi i k n 0 . math Euler s identity is the case where math n 2. In another field of mathematics ... sup 2 sup 1, and pi is pi , the ratio of the circumference of a circle to its diameter. Euler s Identity is also sometimes called Euler s Equation . Mathematical beauty Euler s identity is considered ... exactly once each addition , multiplication , and exponentiation . The identity also links five fundamental mathematical constant s The 0 number number 0 , the additive identity. The 1 number number 1 , the multiplicative identity. The pi number pi , which is ubiquitous in trigonometry , the geometry of Euclidean space , and mathematical analysis analytical mathematics pi 3.14159265... The e mathematical ... unit number math i , the imaginary unit of the complex number s, a field mathematics field of numbers ... . Furthermore, in algebra and other areas of mathematics, equation s are commonly written with zero ... s Identity tied with Maxwell s equations of electromagnetism as the greatest equation ever . ref Crease, 2004. ref An entire 400 page mathematics book, Dr. Euler s Fabulous Formula published in 2006 ... s Identity. This monograph states that Euler s Identity sets the gold standard for mathematical beauty. ref Cited in Crease, 2007. ref Constance Reid claimed that Euler s Identity was the most famous formula in all mathematics. ref Reid, http books.google.com books?id d3NFIvrTk4sC&pg PA155 p ... class mathematician. ref Derbyshire p.210. ref After proving Euler s Identity during a lecture ... Ad8hAx 6m9oC&pg PA103 p.103 104 . ref Stanford University mathematics professor Dr. Keith Devlin said ... p.1 . ref Explanation File Euler s formula.svg thumb right Euler s formula for a general angle The identity ... gives the identity math e i pi 1 0. , math Generalizations Euler s Identity is also a special case ... more details
in this case, the equity . In most cases, each component of an accounting identity can be broken down into further sub groups that must also respect the identity. This usage of the term identity is similar to the mathematical definition of an identitymathematicsidentity . Identities in accounting ...In finance and economics , an accounting identity is an equality that must be true regardless of the value of its variables, or a statement that by definition or construction must be true. ref Principles of Macroeconomics , Mankiw et al., pp. 211 212, 2002 ref ref Macroeconomics Canadian Edition , Mankiw ... or construction true, such as the balance of payments . Where an accounting identity applies, any deviation from the identity signifies an error in formulation, calculation or measurement ... It is important to note that this relationship is an accounting identity. This means that the relationship must be true as long as all variables are measured properly. ref The term accounting identity ... Theory and Policy In practice, this identity rarely adds up, however, because the variables are not typically measured accurately. ref Description The most basic identity in accounting is that the balance ... liabilities. Because this accounting identity must always hold, any change to one side of the equation ... identity, ref Money, Banking, and Financial Institutions , Siklos, pp.145 147, 2006 ref where Current ... identity is that, due to measurement error, the balance of payments may not total correctly ..., 2000. ref Gross domestic product The basic equation for gross domestic product is also considered an identity, and is sometimes referred to as the National Income Identity ref Macroeconomics , Auerbach ... export s International trade import s See also Identity social science Identity Accounting Double entry accounting General Du Pont Identity Business Income statement , Cash flow statement , Balance ... BasicAccountingIdentity.htm The Basic Accounting Identity Category Financial accounting Category Financial ... more details
In linear algebra , the identity matrix or unit matrix of size n is the n n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by I sub n sub , or simply by I if the size is immaterial or can be trivially determined by the context. In some fields, such as quantum mechanics , the identity matrix is denoted by a boldface one, 1 otherwise it is identical to I . math I 1 begin bmatrix 1 end bmatrix , I 2 begin bmatrix 1 & 0 0 & 1 end bmatrix , I 3 begin bmatrix 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 end bmatrix , cdots , I n begin bmatrix 1 & 0 & cdots & 0 0 & 1 & cdots & 0 vdots & vdots & ddots & vdots 0 & 0 & cdots & 1 end bmatrix math Some mathematics books use U and E to represent the Identity Matrix meaning Unit Matrix and Elementary Matrix , or from the German Einheitsmatrix , ref Identity Matrix On Wolfram s MathWorld http mathworld.wolfram.com IdentityMatrix.html ref ... multiplication that math I mA AI n A. , math In particular, the identity matrix serves as the unit of the ring mathematics ring of all n n matrices, and as the identity element of the general linear group GL n consisting of all invertible matrix invertible n n matrices. The identity matrix itself ... s from an n dimensional vector space to itself, I sub n sub represents the identity function , regardless of the Basis linear algebra basis . The i th column of an identity matrix is the unit vector e sub i sub . It follows that the determinant of the identity matrix is 1 and the trace linear ... using the Kronecker delta notation math I n ij delta ij . , math The identity matrix also has the property ... of one another. The identity matrix of a given size is the only idempotent matrix of that size ... . The Square root of a matrix principal square root of an identity matrix is itself, and this is its only Positive definite matrix positive definite square root. However, every identity matrix with at least ... External links planetmath reference title Identity matrix id 1223 Category Abstract algebra Category ... more details
In mathematics , the cyclotomic identity states that math 1 over 1 alpha z prod j 1 infty left 1 over 1 z j right M alpha,j math where M is Moreau s necklace counting function , math M alpha,n 1 over n sum d , ,n mu left n over d right alpha d, math and is the classic M bius function of number theory . The name comes from the denominator, 1 &minus z sup j sup , which is the product of cyclotomic polynomial s. The left hand side of the cyclotomic identity is the generating function for the free associative algebra on generators, and the right hand side is the generating function for the universal enveloping algebra of the free Lie algebra on generators. The cyclotomic identity witnesses the fact that these two algebras are isomorphic. There is also a symmetric generalization of the cyclotomic identity found by Strehl math prod j 1 infty left 1 over 1 alpha z j right M beta,j prod j 1 infty left 1 over 1 beta z j right M alpha,j math References Citation last1 Metropolis first1 N. last2 Rota first2 Gian Carlo author2 link Gian Carlo Rota editor1 last Greene editor1 first Curtis title Combinatorics and algebra Boulder, Colo., 1983 . Proceedings of the AMS IMS SIAM joint summer research conference held at the University of Colorado, Boulder, Colo., June 5 11, 1983. url http books.google.com books?id 2axt00oBDEwC&pg PA19 publisher American Mathematical Society location Providence, R.I. series Contemp. Math. isbn 978 0 8218 5029 9 mr 777692 year 1984 volume 34 chapter The cyclotomic identity pages 19 27 Category Mathematical identities ... more details
In mathematics , in the field of ordinary differential equation s, the Picone identity , named after Mauro Picone , is a classical result about homogeneous differential equation homogeneous linear second order differential equations. It is useful in studying the oscillation differential equation oscillation of such equations and has been generalized to other type of differential equation s and difference equation s. The Picone identity is used to prove the Sturm Picone comparison theorem . Picone identity Suppose that u and v are solutions of the two homogeneous linear second order differential equations in self adjoint form math p 1 x u q 1 x u 0 , math and math p 2 x v q 2 x v 0. , math Then, for all x with v x 0, the following identity holds math left frac u v p 1 u v p 2 u v right left q 2 q 1 right u 2 left p 1 p 2 right u 2 p 2 left u v frac u v right 2. math References cite journal last Picone first Mauro authorlink Mauro Picone title Sui valori eccezionali di un parametro da cui dipende un equazione differenziale lineare del secondo ordine journal Ann. Scuola Norm. Sup. Pisa volume 11 pages 1 141 year 1910 cite journal last Swanson first Charles A. title Picone s Identity journal Rendiconti di Matematica volume 8 issue 2 pages 373 397 year 1975 Category Ordinary differential equations Category Mathematical identities it Identit di Picone ... more details
The Bourne Identity may refer to The Bourne Identity novel The Bourne Identity novel , a 1980 novel by Robert Ludlum The Bourne Identity 1988 film The Bourne Identity 1988 film , a telefilm adaptation of the novel, starring Richard Chamberlain and Jaclyn Smith The Bourne Identity 2002 film The Bourne Identity 2002 film , a film adaptation of the novel, starring Matt Damon See also The Born Identity , an episode of Ugly Betty disambig fr La M moire dans la peau nl The Bourne Identity nds The Bourne Identity ... more details
Internet identity may refer to Online identity personal self concept as it relates to the Internet cf. Identity social science Digital identity Note possibly WP MERGE redundant with above Identity 2.0 technology related to helping identification on the Internet Internet Identity , a company disambig ... more details
Double Identity may refer to Double Identity novel Double Identity novel , a 2005 young adult novel by Margaret Peterson Haddix Double Identity 2009 film Double Identity 2009 film , originally titled Fake Identity , a 2009 film starring Val Kilmer Double Identity Gossip Girl Double Identity Gossip Girl , a 2010 episode of Gossip Girl dab ... more details
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