In linear algebra , a Hessenbergmatrix is a special kind of square matrix , one that is almost Triangular matrix triangular . To be exact, an upper Hessenbergmatrix has zero entries below the first diagonal Matrices subdiagonal , and a lower Hessenbergmatrix has zero entries above the first Diagonal ... matrix triangular matrices , and this improvement often carries over to Hessenberg matrices as well. 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. In fact, reduction of any matrix to a Hessenberg form can be achieved in a finite number of steps for example ... . Subsequent reduction of Hessenbergmatrix to a triangular matrix can be achieved through iterative procedures, such as shifted QR decomposition QR factorization. In eigenvalue algorithm s, the Hessenbergmatrix can be further reduced to a triangular matrix through Shifted QR factorization combined with deflation steps. Reducing a general matrix to a Hessenbergmatrix and then reducing further to a triangular ... of a Hessenbergmatrix with a triangular matrix is again Hessenberg. More precisely, if A is upper Hessenberg and T is upper triangular, then AT and TA are upper Hessenberg. A matrix that is both upper Hessenberg and lower Hessenberg is a tridiagonal matrix . See also Hessenberg variety Notes references ... HessenbergMatrix.html Hessenbergmatrix at MathWorld. http planetmath.org encyclopedia HessenbergMatrix.html Hessenbergmatrix at PlanetMath. http www.cs.utexas.edu users flame pubs flawn53.pdf ... DEFAULTSORT HessenbergMatrix Category Matrices de Hessenbergmatrix es Matriz de Hessenberg eo Matrico ... 251 ref They are named after Karl Hessenberg . ref Biswa Nath Datta 2010 Numerical Linear Algebra ... bmatrix math is upper Hessenberg and math begin bmatrix 1 & 2 & 0 & 0 5 & 2 & 3 & 0 3 & 4 & 3 & 7 5 & 6 & 1 & 1 end bmatrix math is lower Hessenberg. Computer programming Many linear algebra algorithm ... more details
Hessenberg may refer to People Gerhard Hessenberg 1874 1925 , German mathematician Karl Hessenberg 1904 1959 , German mathematician and engineer Kurt Hessenberg 1908 1994 , German composer and professor at the Hochschule f r Musik und Darstellende Kunst in Frankfurt am Main Mathematics Hessenbergmatrix , one that is almost triangular Hessenberg variety , a family of subvarieties of the full flag variety which are defined by a Hessenberg function h and a linear transformation X disambiguation de Hessenberg ... more details
Karl Adolf Hessenberg September 8, 1904 &ndash February 22, 1959 was a Germans German mathematician and engineer . The Hessenbergmatrix form is named after him. However, Ordinal arithmetic Natural operations Hessenberg sum and product of ordinals are named for the mathematician Gerhard Hessenberg , a near relative. Hessenberg was also the brother of composer Kurt Hessenberg , and the great grandson of doctor and author Heinrich Hoffmann author Heinrich Hoffmann . External links http www.hessenberg.de karl1.html Biography of Hessenberg by his daughter, Brigitte Bossert MathGenealogy id 65423 Persondata Metadata see Wikipedia Persondata . NAME Hessenberg, Karl A. ALTERNATIVE NAMES SHORT DESCRIPTION DATE OF BIRTH September 8, 1904 PLACE OF BIRTH DATE OF DEATH February 22, 1959 PLACE OF DEATH DEFAULTSORT Hessenberg, Karl A. Category 1904 births Category 1959 deaths Category 20th century mathematicians Category German mathematicians Category Numerical analysts Germany mathematician stub de Karl Hessenberg it Karl Hessenberg ht Karl Hessenberg pt Karl Hessenberg ... more details
Gerhard Hessenberg Frankfurt , 16 August 1874 Berlin , 16 November 1925 was a Germany German mathematician . He received his Ph.D from the University of Berlin in 1899 under the guidance of Hermann Schwarz and Lazarus Fuchs . His name is usually associated with projective geometry , where he is known for proving that Desargues theorem is a consequence of Pascal s theorem , ref http www.cirm.univ mrs.fr web.ang liste rencontre Resumes05 Rencontre22 Rencontre22.htm Hessenberg s Work in Projective Geometry at www.cirm.univ mrs.fr ref and set theory where he is known for introducing the concept of a Connection mathematics connection . ref http scienceworld.wolfram.com biography Hessenberg.html Hessenberg, Gerhard 1874 1925 from Eric Weisstein s World of Scientific Biography ref The Ordinal arithmetic Natural operations Hessenberg sum and product of ordinals are named for him. However, HessenbergmatrixHessenberg matrices are named for Karl Hessenberg , a near relative. Notes reflist External links http www.genealogy.math.ndsu.nodak.edu html id.phtml?id 45119 Gerhard Hessenberg at the Mathematics Genealogy Project. Persondata Metadata see Wikipedia Persondata . NAME Hessenberg, Gerhard ALTERNATIVE NAMES SHORT DESCRIPTION DATE OF BIRTH 16 August 1874 PLACE OF BIRTH DATE OF DEATH 16 November 1925 PLACE OF DEATH DEFAULTSORT Hessenberg, Gerhard Category 1874 births Category 1925 deaths Category German mathematicians Category 19th century mathematicians Category Geometers Germany mathematician stub de Gerhard Hessenberg fr Gerhard Hessenberg ht Gerhard Hessenberg pms Gerhard Hessenberg pl Gerhard Hessenberg pt Gerhard Hessenberg ... more details
Kurt Hessenberg August 17, 1908 June 17, 1994 was a German composer and professor at the Hochschule f r ... am Main . Life Kurt Hessenberg was born on August 17, 1908 in Frankfurt am Main , Germany , as the fourth and last child of the lawyer Eduard Hessenberg and his wife Emma, n e Kugler. Among his ... Hessenberg was to arrange for children s choir op. 49 later in his life. From 1927&ndash 1931 Hessenberg ... composition and Robert Teichm ller piano . In 1933 Hessenberg became a teacher at the Hoch Conservatory ... lessons. In 1940 Hessenberg received the Nationaler Kompositionspreis national prize for composition ... Vom Wesen und Vergehen op. 45. Hessenberg was appointed professor of composition at the Hochschule f r Musik und Darstellende Kunst in 1953 and taught there until his retirement in 1973. Kurt Hessenberg ... Autobiography , which Hessenberg himself wrote in the late 1980s Hessenberg 1990a . Information on awards and prizes, Hessenberg s career, as well as the date of death are taken from Kirchberg 2001. ref Hessenberg s work contributed significantly to the repertoire of the Protestant churches in the 20th ... Citation last Kirchberg first Klaus contribution Hessenberg, Kurt editor last Sadie editor first ... publisher Macmillan place London year 2001 . Citation first Kurt last Hessenberg editor last Cahn editor first Peter contribution Kleine Selbstbiographie title Kurt Hessenberg Beitr ge zu Leben und Werk ... publisher Cassandra Records accessdate 2008 02 27 Further reading Catalogues of Hessenberg s works Citation first Kurt last Hessenberg title Werkverzeichnis publisher Schott location Mainz year 1968 citation first Kurt last Hessenberg editor last Cahn editor first Peter contribution Werkverzeichnis Kurt Hessenberg title Kurt Hessenberg Beitr ge zu Leben und Werk publisher Schott place Mainz ... halte Kurt Hessenberg geboren 17.8.1908 title Credo musicale Komponistenportraits aus der Arbeit des ... editor last Cahn editor first Peter title Kurt Hessenberg Beitr ge zu Leben und Werk publisher ... more details
In geometry , Hessenberg varieties , first studied by De Mari, Claudio Procesi Procesi , and Shayman, are a family of subvarieties of the full flag variety which are defined by a Hessenberg function h and a linear transformation X . The study of Hessenberg varieties was first motivated by questions in numerical analysis in relation to algorithms for computing eigenvalues and eigenspaces of the linear operator X . Later work by Springer, Peterson, Kostant, among others, found connections with combinatorics , representation theory and cohomology . Definitions A Hessenberg function is a function of tuples math h 1,2, ldots,n rightarrow 1,2, ldots,n math where math h i 1 geq text max i,h i text for all 1 leq i leq n 1. math For example, math h 1,2,3,4,5 2,3,3,4,5 , math is a Hessenberg function. For any Hessenberg function h and a linear transformation math X C n rightarrow C n, , math the Hessenberg variety is the set of all flags math F bullet math such that math X cdot F i subseteq F h i math for all i. Here math F h i math denotes the vector space spanned by the first math h i math vectors in the flag math F bullet math . math mathcal H X,h F bullet mid X F i subset F h i text for 1 leq i leq n math Examples Some examples of Hessenberg varieties with their math h math function include The Full Flag variety h i n for all i The Peterson variety math h i i 1 math for math i 1,2, dots, n 1 math The Springer variety math h i i math for all math i math . References Reflist F. De Mari, Claudio Procesi C. Procesi , and M. Shayman, Hessenberg varieties , Trans. Amer. Math. Soc. 332 1992 , 529 534. B. Kostant , Flag Manifold Quantum Cohomology , the Toda Lattice, and the Representation with Highest Weight math rho math , Selecta Mathematica. N.S. 2 , 1996, 43&ndash 91. J. Tymoczko, Linear conditions imposed on flag varieties , Amer. J. Math. 128 2006 , 1587 1604. Category Algebraic geometry Category Algebraic combinatorics Category Article Feedback 5 ... more details
wiktionary Matrix may refer to TOCright Math and science Matrix mathematics , a rectangular array of numbers, symbols or expressions Matrix biology , the material or tissue between cells in which more specialized structures are embedded Matrix chemical analysis , the non analyte components of a sample Matrix geology , the fine grained material in which larger objects are embedded Harris matrix , an archeological tool Metal matrix , material used in a metal matrix composite Matrix isolation , an experimental technique in chemistry and physics Film and television The Matrix franchise The Matrix franchise , a science fiction franchise The Matrix , the first film in the franchise The Matrix Music from the Motion Picture , a soundtrack album Matrix TV series Matrix TV series Matrix Doctor Who Matrix Doctor Who , a computer system Matrix, a ReBoot character John Matrix, a Commando film Commando character Music and audio Matrix music , an unchanging element of variations Matrix artist , a British DJ Matrix jazz group The Matrix production team M rix , an album by P.A.L. Clave matrix Keyboard matrix circuit Matrix decoder , a technology for converting audio channels Matrix or matrix master , a master disc used in the production of gramophone records Matrix , a song by Chick Corea on the album Now He Sings, Now He Sobs Comics and literature Matrix comics , a DC Comics character Matrix Doctor Who novel Matrix Doctor Who novel , by Robert Perry and Mike Tucker Matrix of Leadership , a Transformers plot element Matrix, a Neuromancer plot element Business and law MATRIX Northern Ireland Science Industry Panel Matrix Business Technologies , a telecommunications firm Matrix Chambers , a barristers set Matrix Games , an American video game publisher founded 1999 Matrix management , an organizational management style Matrix Partners , a private equity investment firm Matrix Software , a Japanese video game developer founded 1994 Growth share matrix , a business analysis technique Transportation ... more details
About the 1999 film the franchise it initiated The Matrix franchise other uses Infobox film name The Matrix image The Matrix Poster.jpg caption Theatrical release poster director ubl The Wachowskis Andy ... name boxoffice The Matrix is a 1999 American science fiction film science fiction action film written ... sequels, and the The Matrix franchise Matrix franchise was further expanded through the production ... source. Upon learning this, computer programmer Neo The Matrix Neo is drawn into a rebellion against ... detail to this summary. Computer programmer Neo The Matrix Thomas Anderson Keanu Reeves is secretly ... references to the Matrix appearing on his computer. A female hacker named Trinity The Matrix Trinity Carrie Anne Moss confirms that a man named Morpheus The Matrix Morpheus Laurence Fishburne can help him however, three sinister Agent The Matrix Agents , led by Agent Smith Hugo Weaving , arrest ... energy of humans, who are kept docile within the Matrix, a simulated reality of the world as it was in 1999. Morpheus and his crew belong to a group of free humans who unplug others from the Matrix ... within the Matrix by using their understanding of its true nature to manipulate its physical law s. Neo is warned that fatal injuries within the Matrix will also kill one s physical body, and that the Agents ... the Matrix to visit the Oracle The Matrix Oracle Gloria Foster , who predicted the emergence of the One ... escape. Their ally Cypher The Matrix Cypher Joe Pantoliano had betrayed them, making a deal for a permanent return to the comfortable life of the Matrix in exchange for giving Morpheus to the Agents. Cypher murders three crew members before he is killed by Tank The Matrix Tank Marcus Chong . The Agents ... in Zion The Matrix Zion , the humans subterranean refuge in the real world. Neo and Trinity return to the Matrix and rescue their leader in the process, Neo becomes more confident in his ability to manipulate the Matrix, ultimately dodging bullets fired at him. Morpheus and Trinity use a telephone ... more details
In mathematics , a bidiagonal matrix is a matrix mathematics matrix with non zero entries along the main diagonal and either the diagonal above or the diagonal below. This means there are exactly two non zero diagonals in the matrix. When the diagonal above the main diagonal has the non zero entries the matrix is upper bidiagonal . When the diagonal below the main diagonal has the non zero entries the matrix is lower bidiagonal . For example, the following matrix is upper bidiagonal math begin pmatrix 1 & 4 & 0 & 0 0 & 4 & 1 & 0 0 & 0 & 3 & 4 0 & 0 & 0 & 3 end pmatrix math and the following matrix is lower bidiagonal math begin pmatrix 1 & 0 & 0 & 0 2 & 4 & 0 & 0 0 & 3 & 3 & 0 0 & 0 & 4 & 3 end pmatrix . math Usage One variant of the QR algorithm starts with reducing a general matrix into a bidiagonal one. ref Bochkanov Sergey Anatolyevich. ALGLIB User Guide General Matrix operations Singular value decomposition . ALGLIB Project. 2010 12 11. URL http www.alglib.net matrixops general svd.php. Accessed 2010 12 11. Archived by WebCite at http www.webcitation.org 5utO4iSnR ref and the Singular value decomposition uses this method as well. See also Diagonal matrix List of matrices LAPACK Bidiagonalization Hessenberg form The Hessenberg form is similar, but has more non zero diagonal lines than 2. Tridiagonal matrix with three diagonals References Stewart, G. W. 2001 Matrix Algorithms, Volume II Eigensystems . Society for Industrial and Applied Mathematics. ISBN 0 89871 503 2. Reflist External links http www.cs.utexas.edu users flame pubs flawn53.pdf High performance algorithms for reduction to condensed Hessenberg, tridiagonal, bidiagonal form Category Linear algebra Category Sparse matrices Linear algebra stub compu prog stub de Bidiagonalmatrix es Matriz bidiagonal eu Matrize bidiagonal sl Bidiagonalna matrika sv Bidiagonal matris ... more details
. Properties A tridiagonal matrix is of HessenbergmatrixHessenberg type in particular, a tridiagonal matrix is a direct sum of p 1 by 1 and q 2 by 2 matrices such that p q 2 n the dimension of the tridiagonal. Although a general tridiagonal matrix is not necessarily Symmetric matrix symmetric or Hermitian matrix Hermitian , many of those that arise when solving linear algebra problems have one of these properties. Furthermore, if a real tridiagonal matrix A satisfies a sub k , k 1 sub a sub k 1, k sub 0, so that the signs of its entries are symmetric, then it is similar matrix similar to a Hermitian matrix, and hence, its eigenvalue s are real. The latter conclusion continues to hold ...In linear algebra , a tridiagonal matrix is a matrix that has nonzero elements only in the main diagonal ... matrix is tridiagonal math begin pmatrix 1 & 4 & 0 & 0 3 & 4 & 1 & 0 0 & 2 & 3 & 4 0 & 0 & 1 & 3 end pmatrix . math The determinant of a tridiagonal matrix is given by a Continuant mathematics ... to tridiagonal matrices as well. For instance, the determinant of a tridiagonal matrix A of order ... the determinant of a tridiagonal matrix using this formula is linear in n , while the cost is cubic for a general matrix. Computer programming A transformation that reduces a general matrix to Hessenberg form will reduce a Hermitian matrix to tridiagonal form. So, many eigenvalue algorithm s, when applied to a Hermitian matrix, reduce the input Hermitian matrix to tridiagonal form as a first step. A tridiagonal matrix can also be stored more efficiently than a general matrix by using a special matrix representation storage scheme . For instance, the LAPACK Fortran package stores an unsymmetric tridiagonal matrix of order n in three one dimensional arrays, one of length n containing the diagonal ... of tridiagonal matrix math A x b math , for math b in reals n math can be solved by a specific algorithm called Tridiagonal matrix algorithm , requiring O n operations Golub and Van Loan ... more details
triangular matrices . Upper and lower HessenbergmatrixHessenberg matrices . Block diagonal matrix ...citations missing date May 2010 In mathematics , particularly matrix theory , a band matrix is a sparse matrix whose non zero entries are confined to a diagonal band , comprising the main diagonal and zero or more diagonals on either side. Matrix bandwidth Bandwidth sparse matrix , matrix bandwidth , bandwidth matrix , bandwidth matrix theory redirect here Formally, consider an n × n matrix A a sub i,j sub . If all matrix elements are zero outside a diagonally bordered band whose range is determined ... of the matrix is k sub 1 sub k sub 2 sub 1 in other words, it is the smallest number of adjacent diagonals to which the non zero elements are confined . A matrix is called a band matrix or banded matrix if its bandwidth is reasonably small. A band matrix with k sub 1 sub k sub 2 sub 0 is a diagonal matrix a band matrix with k sub 1 sub k sub 2 sub 1 is a tridiagonal matrix when k sub 1 sub k sub 2 sub 2 one has a pentadiagonal matrix and so on. If one puts k sub 1 sub 0, k sub 2 sub n &minus 1, one obtains the definition of an upper triangular matrix similarly, for k sub 1 sub n &minus 1, k sub 2 sub 0 one obtains a lower triangular matrix. Applications In numerical ... a matrix with a half bandwidth equal to the square root of the matrix dimension, but inside the band ... decomposition to such a matrix results in the band being filled in by many non zero elements. Band storage ... zero. For example, a tridiagonal matrix has bandwidth 1. The 6 by 6 matrix math begin bmatrix B 11 & B ... & cdots & 0 & B 65 & B 66 end bmatrix math is stored as the 6 by 3 matrix math begin bmatrix 0 & B ... & 0 end bmatrix . math A further saving is possible when the matrix is symmetric. For example, consider a symmetric 6 by 6 matrix with a right bandwidth of 2 math begin bmatrix A 11 & A 12 & A 13 & 0 ... & sym & & & A 55 & A 56 & & & & & A 66 end bmatrix . math This matrix is stored as the 6 by 3 matrix ... more details
QR decomposition Cholesky decomposition Hessenbergmatrix Tridiagonal matrix Invariant subspace ...dablink A related concept is that of a triangular array . for the rings triangular matrix ring File Cyclic group Z4 Cayley table powers of Gray code permutation small .svg thumb Logical matrix Binary lower unitriangular Toeplitz matrix Toeplitz matrices, multiplied using Finite field F sub 2 sub operations ... of linear algebra , a triangular matrix is a special kind of square matrix square matrix mathematics matrix . A square matrix is called lower triangular if all the entries above the main diagonal are zero. Conversely a square matrix is called upper triangular if all the entries below the main diagonal are zero. A triangular matrix is one that is either lower triangular or upper triangular. A matrix that is both upper and lower triangular is a diagonal matrix . Because matrix equations with triangular ... decomposition algorithm, any invertible matrix may be written as the product of a lower triangular matrix L by an upper triangular matrix U . Description A matrix of the form math L begin bmatrix l 1,1 ... n,2 & ldots & l n,n 1 & l n,n end bmatrix math is called a lower triangular matrix or left triangular matrix , and analogously a matrix of the form math U begin bmatrix u 1,1 & u 1,2 & u 1,3 & ldots ... n,n end bmatrix math is called an upper triangular matrix or right triangular matrix . The variable L standing for lower or left is commonly used to represent a lower triangular matrix, while the variable U standing for upper or R standing for right is commonly used for upper triangular matrix. A matrix that is both upper and lower triangular is diagonal matrix diagonal . Matrices that are similar matrix similar to triangular matrices are called triangularisable . The standard operations on triangular ... upper triangular matrix is upper triangular. The product of an upper triangular matrix by a constant is an upper triangular matrix. Together these facts mean that the upper triangular matrices form ... more details
The Matrix code can refer to A two dimensional barcode as opposed to linear and stacked symbologies , such as e.g. Data Matrix . The logo of The Matrix franchise. See Matrix digital rain . disambig ... more details
Matrix algebra may refer to Matrix theory , is the branch of mathematics that studies matrix mathematics matrices Matrix ring , thought of as an algebra over a field or a commutative ring disambig pl Algebra macierzy ... more details
Regular matrix may refer to regular stochastic matrix , a stochastic matrix such that all the entries of some power of the matrix are positive. e.g. Lewis, Matrix Theory, p. 169 invertible matrix this usage is rare . e.g. Plato et al., Concise Numerical Algebra, p. 60 the opposite of irregular matrix , a matrix with a different number of entries in each row. Sansui QS Regular Matrix system for Quadraphonic sound. mathdab ... more details
Z matrix may mean Z matrix chemistry , a table of the locations of atoms comprising a molecule Z matrix mathematics , a matrix whose off diagonal entries are less than or equal to zero It may also refer to The matrix of Z parameters , a matrix characterizing an electrical network disambig ... more details
Merge from Matrix biology date May 2010 Unreferenced date December 2009 In biology , matrix plural matrices is the material or tissue between animal or plant cell biology cells , in which more specialized structures are embedded, and a specific part of the mitochondrion that is the site of oxidation of organic molecules. The internal structure of connective tissue s is an extracellular matrix . Finger nail s and toenail s grow from matrices. Tissue matrices Extracellular matrix Main Extracellular matrix The molecules forming the glue between cells in connective tissues are summarily referred to as the extracellular matrix. Bone matrix Main Osteon Bone is a form of connective tissue found in the body and has a storage area, or osteon region, also called matrix that allows mineral salts such as calcium to be stored. Subcellular matrices Mitochondrial matrix Main Mitochondrial matrix In the mitochondrion , the matrix contains soluble enzymes that catalysis catalyze the oxidation of pyruvic acid pyruvate and other small Organic chemistry organic molecule s. Nuclear matrix Main Nuclear matrix In the cell nucleus the matrix is the insoluble fraction that remains after extracting the solubled DNA . Golgi matrix The Golgi matrix is a protein scaffold made up of golgin s on the cytoplasm ic side of the Golgi apparatus involved in keeping its shape and membrane stacking. Matrix medium A matrix is also a Growth medium medium in which bacteria are grown cultured . For instance, a Petri dish of agar may be the matrix for culturing a sample swab bed from a patient s throat. See also Matrix biology Matricity Tissues and cells Germinal matrix Hair matrix cell Molecular biology Matrix attachment region Matrix metalloproteinase Matrix protein Bioinformatics and sequence evolution PAM matrix Position specific scoring matrix Similarity matrix Substitution matrix Botany and agriculture Matrix Planting Population biology and ecology Matrix population models DEFAULTSORT Matrix Biology Category ... more details
Data matrix can refer to Matrix mathematics A rectangular array of elements Data matrix statistics A matrix, generally of numbers, representing data used in statistical analysis Data matrix computer An encoded barcode. Data set A collection of data in tabular form. Data matrix multivariate statistics A mathematical Matrix mathematics matrix of data whose rows represent different repetition of an experiment, and whose columns represent different kinds of datum taken for each repetition. dab ... more details
The Carver matrix can refer to CARVER matrix a military based target acquisition system Harris matrix Carver matrix Carver matrix seriation diagram named after Martin Carver which is designed to represent the time lapse in use of recognizable archaeological entities such as floors and pits. disambig ... more details
The term matrix model may refer to one of several concepts In theoretical physics , a matrix model is a system usually a quantum mechanics quantum mechanical system with matrix math matrix valued physical quantity physical quantities . See, for example, Lax pair . The old matrix models are relevant for string theory in two spacetime dimensions. The new matrix model is a synonym for Matrix theory physics Matrix theory . Matrix population models are used to model wildlife and human population dynamics. The Matrix Model of substance abuse treatment was a model developed by the Matrix Institute in the 1980s to treat cocaine and methamphetamine addiction. A concept from Algebraic logic . The term Matrix model is used to describe Matrix management where each worker in an organisation has two managers, one functional managing their expertise, the other executive managing their work on an a project. disambiguation Category Mathematical disambiguation ... more details
In mathematics , an Alexander matrix is a presentation matrix for the Alexander invariant of a knot mathematics knot . See also Alexander polynomial References Refimprove date June 2008 refbegin mathworld title Alexander matrix urlname AlexanderMatrix refend Category Knot theory knottheory stub ... more details
Jacobi matrix may refer to Jacobian matrix and determinant of a smooth map between Euclidean space s or smooth manifold s Jacobi operator Jacobi matrix , a tridiagonal symmetric matrix appearing in the theory of orthogonal polynomials mathdab Long comment to avoid being listed on short pages ... more details
In mathematics , particularly matrix theory , a Stieltjes matrix , named after Thomas Joannes Stieltjes , is a real numbers real symmetric matrix symmetric positive definite matrix with nonpositive off diagonal entries. A Stieltjes matrix is necessarily an M matrix . Every n n Stieltjes matrix is invertible to a nonsingular symmetric nonnegative matrix nonnegative matrix, though the converse of this statement is not true in general for n 2. From the above definition, a Stieltjes matrix is a symmetric invertible Z matrix mathematics Z matrix whose eigenvalues have positive real parts. As it is a Z matrix, its off diagonal entries are less than or equal to zero. See also Hurwitz matrix Metzler matrix References cite book title Iterative Solution of Large Linear Systems author David M. Young publisher Dover Publications date 2003 isbn 0486425487 page 42 cite book title Iterative Methods for Solving Linear Systems author Anne Greenbaum publisher Society for Industrial and Applied Mathematics SIAM date 1987 isbn 089871396X page 162 Numerical linear algebra Category Matrices Category Numerical linear algebra Linear algebra stub sl Stieltjesova matrika ... more details
In mathematics , a moment matrix is a special symmetric square matrix mathematics matrix whose rows and columns are indexed by monomial s. The entries of the matrix depend on the product of the indexing monomials only cf. Hankel matrices . Moment matrices play an important role in polynomial optimization , since positive semidefinite matrix positive semidefinite moment matrices correspond to polynomials which are sums of squares . Definition Empty section date July 2010 See also Empty section date July 2010 External links Category Matrices Linear algebra stub ... more details
In linear algebra , a pentadiagonal matrix is a matrix mathematics matrix that is nearly diagonal matrix diagonal to be exact, it is a matrix in which the only nonzero entries are on the main diagonal, and the first two diagonals above and below it. So it is of the form math begin pmatrix c 1 & d 1 & e 1 & 0 & cdots & cdots & 0 b 1 & c 2 & d 2 & e 2 & ddots & & vdots a 1 & b 2 & ddots & ddots & ddots & ddots & vdots 0 & a 2 & ddots & ddots & ddots & e n 3 & 0 vdots & ddots & ddots & ddots & ddots & d n 2 & e n 2 vdots & & ddots & a n 3 & b n 2 & c n 1 & d n 1 0 & cdots & cdots & 0 & a n 2 & b n 1 & c n end pmatrix . math It follows that a pentadiagonal matrix has at most math 5n 6 math nonzero entries, where n is the size of the matrix. Hence, pentadiagonal matrices are sparse matrix sparse . This makes them useful in numerical analysis . See also tridiagonal matrix PlanetMath attribution id 3927 title Pentadiagonal matrix Category Sparse matrices sl Pentadiagonalna matrika ... more details
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