About relational algebra other uses of the term projectionProjection disambiguation Expert subject Mathematics date February 2009 In relational algebra , a projection is a unary operation written as math pi a 1, ...,a n R math where math a 1,...,a n math is a set of attribute names. The result of such projection is defined as the Set mathematics set obtained when the components of the tuple math R math are restricted to the set math a 1,...,a n math it discards or excludes the other attributes. ref http www.cs.rochester.edu nelson courses csc 173 relations algebra.html ref In practical terms, it can be roughly thought of as picking a sub set of all available columns. For example, if the attributes are name, age , then projection of the relation Alice, 5 , Bob, 8 onto attribute list age yields 5,8 we have discarded the names, and only know what ages are present. Related concepts The closely related concept in set theory see projection set theory differs from that of relational algebra in that, in set ... math 3,7 math onto the second component yields 7. Projection is relational algebra s counterpart ... also Projection set theory References references DEFAULTSORT Projection Relational Algebra Category Relational algebra pt Proje o lgebra relacional ru uk ... with existential quantification, some authorities prefer to define projection in terms ... identical concept occurs in the category of monoid s, called a string projection , which consists ... are the relation math Person math and its projection on some say over the attributes math Age math and math ... projection represents the predicate, There exists Name such that Name is age years old and weighs ... of projection are defined as follows math pi a 1, ...,a n R t a 1,...,a n t in R math where math ... that math t a 1,...,a n a , v a , v in t, a in a 1,...,a n math The result of a projection math pi ... . It is interesting to note that projection over no attributes at all is possible, yielding a relation ... more details
sup 3 sup . Subspaces are a common object of study in linearalgebra. Linearalgebra is a branch ... or linear operators and can be represented by matrix mathematics matrices if a Basis linearalgebra basis is given. The matrix theory is often considered as a part of linearalgebra. Linearalgebra ... analysis . Linearalgebra is central to modern mathematics and its applications. An elementary application of linearalgebra is to find the solution of a system of linear equations in several unknowns ... analysis . Linearalgebra has a concrete representation in analytic geometry and is generalized in operator ... did much to spread the subject of linearalgebra beyond pure mathematics. The origin of many of these ideas ... The main structures of linearalgebra are vector space s and linear map s between them. A vector space ... r F . A fundamental role in linearalgebra is played by the notions of linear combination , span linearalgebra span , and linear independence of vectors and basis linearalgebra basis and the dimension linearalgebra dimension of a vector space. Given a vector space V over a field F , an expression ... are concrete objects. This is a major technique in linearalgebra. This is a potentially useful remark ... of choice AC Every vector space has a basis linearalgebra basis . ref The existence of a basis ... and related topics Since linearalgebra is a successful theory, its methods have been developed ... field of scalars by a ring. In multilinear algebra one considers multivariable linear transformations ... algebra with those of mathematical analysis . See also List of linearalgebra topics Numerical linearalgebra Notes references History Fearnley Sander, Desmond, Hermann Grassmann and the Creation of LinearAlgebra http www.jstor.org stable 2320145 via JSTOR , American Mathematical Monthly 86 1979 ... title LinearAlgebra Done Right publisher Springer edition 2nd isbn 978 0387982588 Citation last Bretscher first Otto date June 28, 2004 title LinearAlgebra with Applications publisher Prentice Hall ... more details
wiktionary projection projector projective tocright Projection , projector , or projective may refer to The display of an image by devices such as Movie projector Video projector Overhead projector Slide projector Camera obscura Map projection , reduces a three dimensional planet to a flat map geography Graphical projection , the production of a two dimensional image of a three dimensional object Parallel projection Orthographic projection , including Plan view Plan , or floor plan view Elevation view Elevation , usually a side view of an exterior Cross section geometry Section , a view of the interior at a particular cutting plane Axonometric projection , including Isometric projection Dimetric projection Trimetric projection Oblique projection , including Cabinet projection Cavalier projection Perspective projection anchor chemistry Chemistry Fischer projection Haworth projection Newman projection Mathematics Projection mathematics , any of several different types of functions, mappings, operations, or transformations 3D projectionProjection relational algebraProjectionlinearalgebra Vector projectionProjection method fluid dynamics Projection set theory Projective geometry is a non Euclidean geometry that involves projective spaces. In another direction, projective module s and projective object s generalize free modules Other Projection alchemy , process in Alchemy Military power projection , the capacity of a state to implement policy by means of force, or the threat thereof, in an area distant from its own territory Psychological projection , or Freudian projection ... or emotions Projection fibers , in neuroscience, white matter fibers that connect the cortex to the lower ... on current statistics and trends See also lookfrom intitle Project disambiguation Projection effect disambiguation disambig de Projektion eo Projekcio fr Projection ko it Proiezione he nl Projectie ja no Projeksjon pl Projekcja pt Proje o ru simple Projection sl Projekcija fi ... more details
Pfaffian ProjectionlinearalgebraProjection 2 2 real matrices Similar matrix Linearalgebra ...This is a list of linearalgebra topics . See also list of matrices glossary of tensor theory . Linear equation s System of linear equations Determinant Minor linearalgebra Minor Cauchy Binet formula Cramer s rule Gaussian elimination Gauss Jordan elimination Strassen algorithm Matrix mathematics Matrices Matrix theory Matrix addition Matrix multiplication Basis transformation matrix Characteristic polynomial Trace matrix Trace Eigenvalue, eigenvector and eigenspace Cayley Hamilton theorem Spread ... Dual space Linear function Linear functional Orthogonality Orthogonal complement Orthogonal projection Outer product Improper rotation Category of vector spaces Linear subspace Euclidean subspace Normed vector space Inner product space Multilinear algebra Tensor Classical treatment of tensors Intermediate treatment of tensors Component free treatment of tensors Tensor algebra Exterior algebra Symmetric algebra Clifford algebra Geometric algebra Affine space Affine transformation Affine group Affine ... related lists Linearalgebra Category Linearalgebra Category Indexes of articles Linearalgebra it Progetto Matematica Elenco di voci sull algebra lineare ... Adjugate Transpose Dot product Symmetric matrix Matrix congruence Congruence relation Linearalgebra Congruence relation Orthogonal matrix Skew symmetric matrix Conjugate transpose Unitary matrix ... decomposition Schur complement Computations Householder transformation Least squares , linear least squares Gram Schmidt process vector geometry Vector s Scalar multiplication Linear combination Linear span Linear independence Basis linearalgebra Basis Coordinates vector Vector space s Basis linearalgebra Basis Change of basis Hamel basis Dimension theorem for vector spaces Hamel dimension Examples of vector spaces Linear map Shear mapping or Galilean transformation Squeeze mapping or Lorentz ... more details
using trace . The trace of a projection i.e., P sup 2 sup P is the rank linearalgebra rank of the projection. The trace of a nilpotent matrix is zero. The product of a symmetric matrix ...In linearalgebra , the trace of an n by n square matrix A is defined to be the sum of the elements on the main ... trace function. Trace of a linear operator Given some linear map f V V V is a finite dimension linearalgebra dimensional vector space generally, we can define the trace of this map by considering ... linearalgebra basis for V and describing f as a matrix relative to this basis, and taking the trace ... , which is the Lie algebra of the special linear group of matrices with determinant 1. The special linear group consists of the matrices which do not change volume, while the special linearalgebra ... LinearAlgebra Category Linearalgebra Category Matrix theory ar ca Tra a d una matriu ... algebra ja pl lad macierzy pt Tra o lgebra linear ru sk Stopa matice ... can be used to define the trace for a linear operator in general. Note that the trace is only defined ... Let T be a linear operator represented by the matrix math begin bmatrix 2&2& 3 1& 1& 3 2 &0 & 1 ... The trace is a linear operator linear map . That is, math mathrm tr A B mathrm tr A mathrm tr B , math ... characterize the trace completely in the sense as follows. Let math f math be a linear functional on the space ... is trivial it is an abelian Lie algebra . In particular, using similarity invariance, it follows that the identity ... n math is a semisimple Lie algebra and thus every element in it is the commutator of some pair of elements, otherwise the derived algebra would be a proper ideal. ref Moreover, any square matrix with zero ... nm math additions. The trace of a projection matrix is the dimension of the target space. math ... for the trace of a linear map. Such a definition can be given using the natural isomorphism canonical isomorphism between the space End V of linear maps on V and V V sup sup , where V sup sup ... more details
In linearalgebra , a linear cone is a subset of a vector space that is closed mathematics closed under multiplication by positive scalar mathematics scalars . In other words, a subset C of a real vector space V is a cone if and only if x belongs to C for any x in C and any positive scalar of V or, more succinctly, if and only if C C for any positive scalar . A cone is said to be pointed if it includes the null vector vector space null vector origin mathematics origin 0 otherwise it is said to be blunt . Some authors use non negative instead of positive in this definition of cone , which restricts the term to the pointed cones only. The definition makes sense for any vector space V which allows the notion of positive scalar i.e., where the ground field is an ordered field , such as spaces over the rational number rational , real algebraic number algebraic , or most commonly real number s. The concept can also be extended for any vector space V whose scalar field is a superset of those fields such as the complex number s, quaternion s, etc. , to the extent that such a space can be viewed as a real vector space of higher dimension. Related concepts The cone of a set The linear cone of an arbitrary subset X of V is the set X sup sup of all vectors x where x belongs to X and ... cone, or a cone that is contained in an open halfspace of V . Properties Boolean, additive and linear closure Linear cones are closed under Boolean operation s Intersection set theory set intersection ... are cones, so is C D and arbitrary linear map s. In particular, if C is a cone, so is its opposite cone C . Spherical section and projection Let be any norm mathematics norm for V , with the property ... by its central projection onto S that is, by the set math C bigg , frac x x x in C wedge x neq ... cones and subsets of S . Indeed, the central projection C is simply the spherical section ... Algebra fr C ne analyse convexe it Cono algebra lineare ... more details
In linearalgebra , subspace may refer to Euclidean subspace , a set of vectors in n dimensional Euclidean space that is closed under addition and scalar multiplication. Linear subspace , the corresponding notion for abstract vector space s. mathdab ... more details
This article collects the main theorems and definitions in linearalgebra . Vector spaces A vector space or linear space V over a number field F consists of a set on which two operations called addition ... , then math s A u math . Canonical forms Empty section date July 2010 References LinearAlgebra 4th edition, by Stephen H. Friedberg Arnold J. Insel and Lawrence E. spence ISBN7040167336 LinearAlgebra 3rd edition, by Serge Lang UTM ISBN0387964126 DEFAULTSORT Theorems And Definitions In LinearAlgebra Category Linearalgebra ... Empty section date February 2011 Linear combinations Empty section date February 2011 Systems of linear equations Empty section date February 2011 Linear dependence Empty section date February 2011 Linear independence Empty section date February 2011 Bases Empty section date February 2011 Dimension Empty section date February 2011 Linear transformations and matrices Linear transformations Null spaces Ranges The matrix representation of a linear transformation Composition of linear transformations ... Dominance relation br Identity matrix br Identity transformation br Incidence matrix br Inverse of a linear transformation br Inverse of a matrix br Invertible linear transformation br Isomorphic vector spaces br Isomorphism br Kronecker delta br Left multiplication transformation br Linear operator br Linear transformation br Matrix representing a linear transformation br Nullity of a linear transformation br Null space br Ordered basis br Product of matrices br Projection on a subspace br Projection on the x axis br Range br Rank of a linear transformation br Reflection about the x axis br ... math N T &R T are subspaces Theorem 2.1 Let V and W be vector spaces and I V W be linear. Then N T and R ... 2.2 Let V and W be vector spaces, and let T V W be linear. If math beta v 1,v 2,...,v n math is a basis ... be linear. If V is finite dimensional, then br math mbox nullity T mbox rank T dim V . math math color ... more details
Numerical linearalgebra is the study of algorithms for performing linearalgebra computations, most notably Matrix mathematics matrix operations, on computer s. It is often a fundamental part of engineering and computational science problems, such as image processing image and signal processing , Telecommunication , computational finance , materials science simulations, structural biology , data mining , and bioinformatics , fluid dynamics , and many other areas. Such software relies heavily on the development, analysis, and implementation of state of the art algorithms for solving various numerical linearalgebra problems, in large part because of the role of matrices in finite difference method finite difference and finite element method s. Common problems in numerical linearalgebra include computing the following LU decomposition , QR decomposition , Singular value decomposition , eigenvalue s. See also Numerical analysis , of which numerical linearalgebra is a subspecialty Gaussian elimination , an important algorithm in numerical linearalgebra BLAS and LAPACK , highly optimized computer libraries which implement most basic algorithms in numerical linearalgebra. List of numerical analysis software List of numerical libraries References cite book last Leader first Jeffery J. title Numerical Analysis and Scientific Computation year 2004 publisher Addison Wesley location isbn 0 201 73499 0 Cite book last1 Bau III first1 David last2 Trefethen first2 Lloyd N. author2 link Lloyd Nicholas Trefethen title Numerical linearalgebra publisher Society for Industrial and Applied ... J. H. Wilkinson and C. Reinsch, LinearAlgebra, volume II of Handbook for Automatic Computation SIAM ... algebra on the web , composed by Jack Dongarra and Hatem Ltaief, University of Tennessee mathapplied stub Category Numerical linearalgebra de Numerische lineare Algebra es lgebra lineal num rica fa id Aljabar linear numerik zh ... more details
theory of cyclic groups. math X 2 X math Projectionlinearalgebra projections are diagonalizable ... . math li li Given a basis linearalgebra basis v sub 1 sub ,..., v sub n sub the minimal polynomial ... , namely the fact that the minimal polynomial is determined by the relations of linear ... a field F if and only if its minimal polynomial factors completely over F into distinct linear factors ... is a linear combination of the first three, in fact math T 3 cdot e 1 4T 2 cdot e 1 T cdot e 1 ... ca Polinomi m nim cs Minim ln polynom line rn algebra de Minimalpolynom el ... more details
Basic LinearAlgebra Subprograms BLAS is a De facto standard de facto application programming interface standard for publishing libraries to perform basic linearalgebra operations such as Vector space vector and matrix multiplication . They were first published in 1979, and are used to build larger packages such as LAPACK . Heavily used in high performance computing , highly optimized implementations of the BLAS interface have been developed by hardware vendors such as Intel and Advanced Micro Devices AMD , as well as by other authors, e.g. Kazushige Goto Goto BLAS and Automatically Tuned LinearAlgebra Software ATLAS a portable self optimizing BLAS . The LINPACK benchmark relies heavily on General Matrix Multiply DGEMM , a BLAS subroutine, for its performance. Functionality The BLAS functionality is divided into three levels 1, 2 and 3. Level 1 This level contains vector operations of the form math boldsymbol y leftarrow alpha boldsymbol x boldsymbol y math as well as scalar dot product s and norm mathematics vector norm s, among other things. Level 2 This level contains matrix vector ... Windows Windows . http developer.amd.com acml.aspx ATLAS Automatically Tuned LinearAlgebra ... implementation of dense linearalgebra library http z.cs.utexas.edu wiki flame.wiki FrontPage MTL4 ... Armadillo is a C linearalgebra library aiming towards a good balance between speed and ease of use ... Sparse extensions to the previously dense BLAS exist such as in ACML See also Numerical linearalgebra , the type of problem BLAS solves LAPACK , the LinearAlgebra Package Math Kernel Library List ... LinearAlgebra Subprograms The New Standard from the BLAS Technical Forum doi inline 10.1145 567806.567810 Numerical linearalgebra Category Numerical linearalgebra Category Numerical software de Basic LinearAlgebra Subprograms fr Basic LinearAlgebra Subprograms ja Basic LinearAlgebra Subprograms pl Basic LinearAlgebra Subprograms ru Basic LinearAlgebra Subprograms uk Basic LinearAlgebra ... more details
unreferenced date June 2010 The following tables provide a comparison of linearalgebra Library computing software libraries , either specialized or general purpose libraries with significant linearalgebra coverage. Dense linearalgebra General information class wikitable sortable style font size smaller text align center width auto style width 12em Creator Language First public release Latest stable version Cost United States dollar USD License Notes ALGLIB Sergey Bochkanov C , C , FreePascal, VBA 2006 2.6.0 06.2010 free GPL General purpose numerical analysis library. Uses automatic code generation to support several programming languages. Automatically Tuned LinearAlgebra Software ATLAS R. Clint Whaley et al. C 2001 3.8.3 02.2009 free BSD Automatically tuned implementation of BLAS. Also includes LU and Cholesky decompositions. GNU Scientific Library GNU Project C 1996 1.14 03.2010 free GNU General Public License GPL General purpose numerical analysis library. Includes some support for linearalgebra. IMSL Numerical Libraries Rogue Wave Software C, Java, C , Fortran, Python 1970 many components nonfree Proprietary General purpose numerical analysis library. LAPACK Fortran late 1980s 3.2.1 04.2009 free BSD licenses 3 clause BSD Numerical linearalgebra library with long history Math Kernel Library MKL Intel C , Fortran 05.2003 10.2 nonfree Proprietary Numerical analysis library optimized for Intel CPU s NAG Numerical Libraries The Numerical Algorithms Group Ltd C, Fortran 1970s many components nonfree Proprietary General purpose numerical analysis library. NMath CenterSpace Software C 2003 4.0 November 2009 nonfree Proprietary Math and statistical libraries for the .NET Framework SciPy Enthought Python 0.8.0 2010 free BSD licenses BSD based on Python Matrix types and operations Matrix types special types like bidiagonal tridiagonal are not listed Real general nonsymmetric ... SVD GEVD GSVD ALGLIB yes yes yes yes no no no yes yes yes yes yes no Automatically Tuned LinearAlgebra ... more details
Unreferenced date December 2009 In mathematics , particularly in linearalgebra , a flag is an increasing sequence of Linear subspace subspaces of a finite dimensional vector space V . Here increasing means each is a proper subspace of the next see filtration abstract algebra filtration math 0 V 0 sub V 1 sub V 2 sub cdots sub V k V. math If we write the dim V sub i sub d sub i sub then we have math 0 d 0 d 1 d 2 cdots d k n, math where n is the dimension linearalgebra dimension of V assumed to be finite dimensional . Hence, we must have k n . A flag is called a complete flag if d sub i sub i , otherwise it is called a partial flag . A partial flag can be obtained from a complete flag by deleting some of the subspaces. Conversely, any partial flag can be completed in many different ways by inserting suitable subspaces. The signature of the flag is the sequence d sub 1 sub , d sub k sub . Bases An ordered basis linearalgebra basis for V is said to be adapted to a flag if the first d sub i sub basis vectors form a basis for V sub i sub for each 0 i k . Standard arguments from linearalgebra can show that any flag has an adapted basis. Any ordered basis gives rise to a complete flag by letting the V sub i sub be the span of the first i basis vectors. For example, the Visible anchor ... Flag LinearAlgebra Category Linearalgebra de Fahne Mathematik fr Drapeau math matiques pl Flaga algebra liniowa ... matrices. More generally, the stabilizer of a flag the linear operators on V such that math T V i V i math for all i is, in matrix terms, the algebra of block upper triangular matrices with respect ... complete flag is a Borel subgroup of the general linear group , and the stabilizer of any partial flags ... order for inclusion and which further is closed under arbitrary intersections and closed linear spans. See nest algebra . Set theoretic analogs See Field with one element From the point of view of the field ... more details
In mathematics , the fundamental theorem of linearalgebra makes several statements regarding vector space s. These may be stated concretely in terms of the rank linearalgebra rank r of an m × n matrix mathematics matrix A and its LU decomposition LDU factorization math PA LDU math wherein P is a permutation matrix , L is a lower triangular matrix , D is a diagonal matrix , and U is an upper triangular matrix . At a more abstract level there is an interpretation that reads it in terms of a linear mapping and its transpose . First, each matrix A induces four fundamental linear subspace subspaces . These fundamental subspaces are table class wikitable tr th name of subspace th definition th containing space th Hamel dimension dimension th basis linearalgebra basis tr td column space , range or image mathematics image td math mathrm im A math or math mathrm range A math td math mathbf R m math td math r math Rank linearalgebra rank td The math r math columns corresponding to those with Pivot element pivots in math mathbf U math tr td nullspace or Kernel mathematics kernel td math mathrm ker A math or math mathrm null A math td math mathbf R n math td math n r math nullity td The math n r math columns of math x math in the solution of math mathbf U mathbf x mathbf 0 math tr td row space or coimage td math mathrm im A T math or math mathrm range A T math td math mathbf R n math td math r math td The math r math rows corresponding to those with pivots in math mathbf U math tr ... theorem References Gilbert Strang Strang, Gilbert . LinearAlgebra and Its Applications . 3rd ed. Orlando Saunders, 1988. Citation title The fundamental theorem of linearalgebra url http www.eng.iastate.edu ... OcwWeb Mathematics 18 06Spring 2005 VideoLectures detail lecture10.htm MIT LinearAlgebra Lecture ... Category Linearalgebra Category Mathematical theorems Category Isomorphism theorems Category Fundamental theorems Linearalgebra ... more details
In linearalgebra , the nonnegative rank of a nonnegative matrix is the smallest number of nonnegative rank one matrices into which the matrix can be decomposed additively p align center math mbox rank A min q mid sum j 1 q R j A, mbox rank ,R 1 dots mbox rank ,R q 1, R 1, dots,R q ge 0 , math p where R sub j sub 0 stands for R sub j sub is nonnegative . Given a nonnegative math m times n math matrix A the nonnegative rank math rank A math of A satisfies p align center math mbox rank , A leq mbox rank A leq min m,n , math p where math rank A math denotes the usual Rank linearalgebra rank of A . Connection with the usual Rank linearalgebra rank It is always true that rank A rank sub sub A . In fact rank sub sub A rank A holds whenever rank A 2 1 . In the case rank A 3 , however, rank A rank sub sub A is possible. For example, the matrix math mathbf A begin bmatrix 1 & 1 & 0 & 0 1 & 0 & 1 & 0 0 & 1 & 0 & 1 0 & 0 & 1 & 1 end bmatrix , math satisfies rank A 3 4 rank sub sub A 1 . Computing the nonnegative rank The nonnegative rank of a matrix can be determined algorithmically, see 1 . It has been proved that determining whether math rank A rank A math is NP hard 3 . Obvious questions concerning the complexity of nonnegative rank computation remain unanswered to date. For example, the complexity of determining the nonnegative matrix of matrices of fixed rank k is unknown for k 2 . Ancillary Facts The minimum number of facets of an extension of a polyhedron P is equal to the nonnegative rank of its so called i slack matrix i . References 1 J. Cohen and U. Rothblum. Nonnegative ranks, decompositions and factorizations of nonnegative matrices . LinearAlgebra and its Applications , 190 149 168, 1993. 2 Abraham Berman and Robert J. Plemmons. Nonnegative Matrices in the Mathematical Sciences , SIAM 3 Stephen Vavasis, On the complexity of nonnegative matrix factorization, SIAM Journal on Optimization 20 3 1364 1377, 2009. math stub Category Linearalgebra ... more details
In mathematics , a frame of a vector space V , is either of two distinct notions, both generalizing the notion of a Basis linear algebra basis In one definition, a k frame k frame is an ordered set of k linearly independent vectors in a space thus k &le n the dimension of the vector space, and if k n an n frame is precisely an ordered basis. If the vectors are orthogonal or orthonormal , the frame is called an orthogonal frame or orthonormal frame , respectively. In the other definition, a frame of a vector space frame is a certain type of ordered set of vectors that spans a space. Thus k &ge n . These are rarely confused and generally clear from context, as the former is a basic concept in finite dimensional geometry, such as Stiefel manifold s, while the latter is most used in analysis. Further, the former must have at most as many elements as the dimension of the space, while the latter must have at least as many elements as the dimension of the space, so the only overlapping sets are bases. See also k frame Frame of a vector space Riemannian geometry Orthonormal frame Moving frame Overcompleteness Category Linear algebra ... more details
http www.netlib.org utk people JackDongarra la sw.html Jack Dongarra s LinearAlgebra Software Page Numerical linearalgebra Category Numerical linearalgebra Category Numerical software Category ... more details
About a concept in linearalgebra the unrelated concept of minor in graph theory Minor graph theory In linearalgebra , a minor of a matrix mathematics matrix A is the determinant of some smaller square matrix , cut down from A by removing one or more of its rows or columns. Minors obtained by removing just one row and one column from square matrices first minors are required for calculating matrix Cofactor linearalgebra cofactor s, which in turn are useful for computing both the determinant and Inverse matrix inverse of square matrices. Detailed definition Let A be an m n matrix and k an integer with 0 k m , and k n . A k k minor of A is the determinant of a k k matrix obtained from A by deleting m k rows and n k columns. Since there are binomial coefficient math m choose k math read Binomial coefficient m choose k ways to choose k rows from m rows, and there are math n choose k math ways to choose k columns from n columns, there are a total of math m choose k cdot n choose k math minors of size k k . Nomenclature The i , j minor often denoted M sub ij sub of an n n square matrix A is defined as the determinant of the n 1 n 1 matrix formed by removing from A its i sup th sup row and j sup th sup column. An i , j minor is also referred to as i , j sup th sup minor, or simply i , j minor. M sub ij sub is also called the minor of the element a sub ij sub of matrix A . A minor that is formed ... or adjoint of a matrix The i , j Cofactor linearalgebra cofactor C sub ij sub of a square ... i j sup M sub ij sub The Cofactor linearalgebra Matrix of cofactors cofactor matrix of A , or matrix ... computed earlier. References reflist linearalgebra Category Matrix theory Category Determinants ar ... algebra approach A more systematic, algebraic treatment of the minor concept is given in multilinear algebra , using the wedge product the k minors of a matrix are the entries in the k th exterior ... it Minore algebra lineare he lv Minors line r algebra nl Minor wiskunde pl Minor ru ... more details
of a matrix is the greatest order of any non zero Minor linearalgebra minor in the matrix the order ... . This proof was adapted from. ref Leonid Mirsky An Introduction to LinearAlgebra , 1990, ISBN 0486664341 ... tensor s necessary to express the matrix as a linear combination, and that this definition does agree with matrix rank as here discussed. See also Rank differential topology Nonnegative rank linearalgebra ... rank linearalgebra Category Linearalgebra cs Hodnost matice de Rang Mathematik es Rango lgebra ...The rank of a matrix mathematics matrix A is the number of linear independence linearly independent rows ... by either rk A or rank A . The rank is the dimension of the image matrix image of the linear transformation that is multiplication by A. More generally, if a linear operator on a vector space ... If one considers the matrix A as a linear mapping T F sup n sup &rarr F sup m sup such that f x A x then the rank of A can also be defined as the dimension of the image of f see linear map for a discussion of image and kernel . This definition has the advantage that it can be applied to any linear map ... algebra kernel of f the rank nullity theorem states that this is the same as the dimension of the image ... of F sup m sup generated by the columns of A , which is in fact just the image of A as a linear ... to a definition of the rank of a linear map the rank of a linear map f from V W is the minimal ... as the tensor rank the minimum number of simple tensor s rank 1 tensors needed to express A as a linear ... either the real numbers or the complex numbers, and we define the linear map f by f x A x as above ... the rank of a matrix is the computation of the number of solutions of a system of linear ... of a matrix can be used to determine whether a linear system is controllability controllable ... map s between smooth manifold s. It is equal to the linear rank of the pushforward differential derivative ... to Matrix Algebra 1. Vectors http numericalmethods.eng.usf.edu mws che 04sle mws che sle bck ... more details
Lie groups In mathematics , the special linear Lie algebra of order n denoted math mathfrak sl n F math is the Lie algebra of math n times n math matrices with trace zero and with the Lie bracket math X,Y XY YX math . This algebra is well studied and understood, and is often used as a model for the study of other Lie algebras. Representation Theory of math mathfrak sl 2 math The simplest non trivial Lie algebra is math mathfrak sl 2 C . math , consisting of two by two matrices with zero trace. There are three basis elements, math e math , math f math , and math h math , with math e left begin array cc 0&1 0&0 end array right math math f left begin array cc 0&0 1&0 end array right math and math h left begin array cc 1&0 0&1 end array right math The commutators are math e,f h math , math h,f 2f math , and math h,e 2e math Let math V math be a finite irreducible Representation mathematics representation of math mathfrak sl 2 math , and let math v math be an eigenvector of math h math with the highest eigenvalue math lambda math . Then, math h,e v hev ehv 2ev math or math hev lambda 2 ev math Since math v math is the eigenvector of highest eigenvalue, math ev 0 math . Similarly, we can show that math hfv lambda 2 fv math and since h has a lowest eigenvalue, there is a math N math such that math f N v 0 math . We will take the smallest math N math such that this happens. We can then recursively calculate math ef k v k 2 lambda 1 k f k 1 v math and we find math e k f k v k h h 1 ... h k 1 v math Taking math k N math , we get math 0 e N f N v N h h 1 ... h N 1 v math Since we chose math N math to be the smallest exponent such that math f N v 0 math , we conclude that math lambda N 1 math . From this, we see that math v math , math fv math , ... math f lambda v math are all nonzero, and it is easy to show that they are linearly independent. Therefore, for each math N math , there is a unique .... Namely, we divide the algebra to three subalgebras h the Cartan algebra Cartan Subalgebra , e , and f ... more details
In linearalgebra , the cofactor sometimes called adjunct , see A remark about different notations below describes a particular construction that is useful for calculating both the determinant and Invertible matrix inverse of square Matrix mathematics matrices . Specifically the cofactor of the i , j entry of a matrix, also known as the i , j cofactor of that matrix, is the Plus and minus signs sign ed Minor linearalgebra minor of that entry. Informal approach to minors and cofactors Finding the minors of a matrix A is a multi step process Choose an entry math a ij math from the matrix. Cross out the entries that lie in the corresponding row math i math and column math j math . Rewrite the matrix without the marked entries. Obtain the determinant math M ij math of this new matrix. math M ij math is termed the minor for entry math a ij math . If i j is an Parity mathematics even number, the cofactor math C ij math of math a ij math coincides with its minor math C ij M ij . , math Otherwise, it is equal to the additive inverse of its minor math C ij M ij . , math Formal definition If A is a square matrix, then the minor of its entry math a ij math , also known as the i , j , or i , j , or i , j sup th sup minor of A , is denoted by math M ij math and is defined to be the determinant of the submatrix obtained by removing from A its i th row and j th column. It follows math C ij 1 i j M ij , math and math C ij math called the cofactor of math a ij math , also referred to as the i , j , i , j or i , j sup th sup cofactor of A . Example Given the matrix math B begin bmatrix b 11 ... also Linearalgebra Matrix mathematics Minor linearalgebra Adjugate matrix References Reflist Citation first1 Howard last1 Anton first2 Chris last2 Rorres title Elementary LinearAlgebra edition ... videoplay?docid 5338528094439680133 MIT LinearAlgebra Lecture on Cofactors at Google Video, from MIT OpenCourseWare http planetmath.org encyclopedia Cofactor.html PlanetMath Category Linearalgebra ... more details
general concept in physics, see frame of reference . In linearalgebra , a basis is a set of linear independence linearly independent vector space vector s that, in a linear combination , can ... , projective space , convex set , and Cone linearalgebra cone have related notions of visible anchor ... link Serge Lang title Linearalgebra publisher Springer Verlag location Berlin, New York isbn ... of linearalgebra 1875 1940 url http www.sciencedirect.com science? ob ArticleURL& udi ... Proof that any subspace basis has same number of elements linearalgebra DEFAULTSORT Basis LinearAlgebra Category Linearalgebra Category Articles containing proofs Category Matroid theory Category ... lin aire ko hr Baza linearna algebra id Basis aljabar linear it Base algebra lineare ... linear ro Baz algebric ru simple Basis linearalgebra sl Baza linearna algebra sr ... field F is a Linear independence linearly independent subset of V that linear span span s or generator ... number s R or C . Then B is a basis if it satisfies the following conditions the linear independence ... infinite linear combinations allow alternative definitions of the basis concept see Related ... , for example, when considering the transformation matrix of a linear map with respect to a basis .... This matrix represents any vector of the new basis as linear combination of the current ... as a linear combination of vectors in B in a unique way. If the basis is ordered see Ordered bases and coordinates Ordered bases and coordinates below then the coefficients in this linear combination ... independent can always span a vector set by expressing it as a linear combination. In P sub 2 ... math a 0. , math Hence we have linear independence. Part II To prove that these two vectors generate ... for V is equivalent to a choice of a linear isomorphism from the coordinate space F sup n sup to V .... Suppose first that &phi F sup n sup V is a linear isomorphism. Define an ordered basis v sub i ... more details
In linearalgebra , the quotient of a vector space V by a linear subspace subspace N is a vector space obtained by collapsing N to zero. The space obtained is called a quotient space and is denoted V N read V mod N . Definition Formally, the construction is as follows harv Halmos 1974 loc 21 22 . Let V be a vector space over a field mathematics field K , and let N be a linear subspace subspace of V . We define an equivalence relation on V by stating that x y if x &minus y &isin N . That is, x is related to y if one can be obtained from the other by adding an element of N . From this definition, one can deduce that any element of N is equivalent to the zero vector in other words all the vectors in N get mapped into the equivalence class of the zero vector. The equivalence class of x is often denoted x x N since it is given by x x n n &isin N . The quotient space V N is then defined as V , the set of all equivalence classes over V by . Scalar multiplication and addition are defined on the equivalence classes by &alpha x &alpha x for all &alpha &isin K , and x y x y . It is not hard to check that these operations are well defined i.e. do not depend on the choice of representative . These operations turn the quotient space V N into a vector space over K with N being the zero class, 0 . The mapping that associates to v &isin V the equivalence class ... isomorphism theorem of linearalgebra says that the quotient space V ker T is isomorphic to the image ... on analysis, Volume II publisher Academic Press year 1970 . Category Linearalgebra Category Functional ... V U given by sending x to its equivalence class x . The kernel algebra kernel or nullspace of this epimorphism ... math mathrm codim U dim V U dim V dim U . math Let T V &rarr W be a linear operator . The kernel of T , denoted ... the rank of T . The cokernel of a linear operator T V &rarr W is defined to be the quotient ... ja pl Przestrze ilorazowa algebra liniowa ru ... more details
In linearalgebra , the coherence ref name tropp cite journal last Tropp first J.A. year 2006 month March title Just relax Convex programming methods for identifying sparse signals in noise journal IEEE Transactions on Information Theory volume 52 issue 3 pages 1030 1051 url http ieeexplore.ieee.org stamp stamp.jsp?arnumber 01603770 doi 10.1109 TIT.2005.864420 ref or mutual coherence ref name donoho06 cite journal last Donoho first D.L. authorlink David Donoho coauthors M. Elad V.N. Temlyakov year 2006 month January title Stable recovery of sparse overcomplete representations in the presence of noise journal IEEE Transactions on Information Theory volume 52 issue 1 pages 6&ndash 18 id url http ieeexplore.ieee.org xpls abs all.jsp?arnumber 1564423 doi 10.1109 TIT.2005.860430 ref of a Matrix mathematics matrix A is defined as the maximum absolute value of the cross correlation s between the columns of A . Formally, let math a 1, ldots, a m math be the columns of the matrix A , which are assumed to be normalized such that math a i H a i 1. math The mutual coherence of A is then defined as ref name donoho06 ref name tropp math M max 1 le i ne j le m left a i H a j right . math The concept was introduced in a slightly less general framework by David Donoho Donoho and Huo, ref cite journal last Donoho first D.L. authorlink David Donoho coauthors Xiaoming Huo year 2001 month November title Uncertainty principles and ideal atomic decomposition journal IEEE Transactions on Information Theory volume 47 issue 7 pages 2845&ndash 2862 url http ieeexplore.ieee.org stamp stamp.jsp?arnumber 00959265 doi 10.1109 18.959265 ref and has since been used extensively in the field of sparse approximation sparse representations of signal electrical engineering signal s. In particular, it is used as a measure of the ability of suboptimal algorithms such as matching pursuit and basis pursuit to correctly ... name tropp See also Compressed sensing Restricted isometry property References reflist algebra stub ... more details
are often epimorphism s and even surjective, they do not have to be. In linearalgebra , a linear ...Image proj map.png thumb right Commutativity of this diagram is the universality of projection , for any map f and set X. In mathematics , a projection is any one of several different types of functions, mappings, operations, or transformations, for example, the following In set theory An operation typified by the j sup th sup projection set theory projection map , written proj sub j sub , that takes an element x x sub 1 sub , ..., x sub j sub , ..., x sub k sub of the cartesian product X sub 1 sub × × X sub j sub × × X sub k sub to the value proj sub j ... equivalence class under a given equivalence relation is known as the canonical projection. The evaluation ... be identified with the cartesian product math prod i in X Y i math , and the evaluation map is a projection ... product of some objects has a canonical projection morphism to each factor. This projection will take many forms in different categories. The projection from the Cartesian product of set mathematics ... x , y , 0 in the plane is a projection. This type of projection naturally generalizes to any number of dimensions n for the source and k n for the target of the mapping. See orthogonal projection , projectionlinearalgebra . In the case of orthogonal projections, the space admits a decomposition as a product, and the projection operator is a projection in that sense as well. In differential topology , any fiber bundle includes a projection map as part of its definition. Locally at least this map looks like a projection map in the sense of the product topology, and is therefore open and surjective ... of the projection map. A retract which is homotopic to the identity is known as a deformation ... resolute scalar projection or resolute of one vector geometric vector onto another. DEFAULTSORT Projection ... ja pl Rzut matematyka pt Proje o matem tica simple Projection mathematics zh ... more details
Encyclopedia
top
