In mathematics and computer science , an adjacencymatrix is a means of representing which Vertex graph ... matrix representation for a graph is the incidence matrix . Specifically, the adjacencymatrix of a finite ... typically use the former convention. There exists a unique adjacencymatrix for each isomorphism class of graphs up to permuting rows and columns , and it is not the adjacencymatrix of any other isomorphism class of graphs. In the special case of a finite simple graph , the adjacencymatrix is a 0,1 matrix with zeros on its diagonal. If the graph is undirected, the adjacencymatrix is symmetric matrix symmetric . The relationship between a graph and the eigenvalue s and eigenvector s of its adjacency ... width 700px height 1100px Labeled graph Adjacencymatrix Image 6n graph2.svg 200px math begin pmatrix ... graph 1,5,21 adjacencymatrix .svg 250px br Coordinates are 0 23. br White fields are zeros, colored ... Cayley graph of Symmetric group S sub 4 sub File Symmetric group 4 Cayley graph 4,9 adjacencymatrix .svg 250px br As the graph is directed, br the matrix is not Symmetric matrix symmetric . The adjacencymatrix of a complete graph is all 1 s except for 0 s on the diagonal. The adjacencymatrix of an empty graph is a zero matrix . Adjacencymatrix of a bipartite graph Adjacencymatrix of a bipartite graph & Biadjacency matrix redirect here The adjacencymatrix A of a bipartite graph whose parts ... Frobenius theorem . Variations An a , b , c adjacencymatrix A of a simple graph has A sub ij sub a if ij is an edge, b if it is not, and c on the diagonal. The Seidel adjacencymatrix is a 1,1,0 adjacencymatrix . This matrix is used in studying strongly regular graph s and two graph s. ref cite journal last Seidel first J. J. title Strongly Regular Graphs with 1,1,0 AdjacencyMatrix Having Eigenvalue ... of edges in it. The distance matrix resembles a high power of the adjacencymatrix, but instead ... structure , the main alternative to the adjacencymatrix is the adjacency list . Because each entry ... more details
In mathematics , in graph theory , the Seidel adjacencymatrix of a simple graph G also called the Seidel matrix and&mdash the original name&mdash the &minus 1,1,0 adjacencymatrix is the symmetric matrix with a row and column for each vertex, having 0 on the diagonal and, in the positions corresponding to vertices v sub i sub and v sub j sub , &minus 1 if the vertices are adjacent and 1 if they are not. The multiset of Eigenvalue eigenvalues of this matrix is called the Seidel spectrum . The Seidel matrix was introduced by van Lint and Seidel 1966 and extensively exploited by Seidel and coauthors. It is the adjacencymatrix of the signed graph signed complete graph in which the edges of G are negative and the edges not in G are positive. It is also the adjacencymatrix of the two graph associated with G . The eigenvalue properties of the Seidel matrix are valuable in the study of strongly regular graph s. See also Adjacencymatrix References van Lint, J.H., and Seidel, J.J. 1966 , Equilateral point sets in elliptic geometry. em Indagationes Mathematicae em , vol. 28 em Proc. Kon. Ned. Aka. Wet. Ser. A em , vol. 69 , pp. 335 348. Seidel, J. J. 1976 , A survey of two graphs. In cite Colloquio Internazionale sulle Teorie Combinatorie cite Proceedings, Rome, 1973 , vol. I, pp. 481 511. Atti dei Convegni Lincei, No. 17. Accademia Nazionale dei Lincei, Rome. Seidel, J.J. 1991 , ed. D.G. Corneil and R. Mathon, cite Geometry and Combinatorics Selected Works of J.J. Seidel cite . Boston Academic Press. Many of the articles involve the Seidel matrix. Seidel, J. J. Strongly Regular Graphs with 1,1,0 AdjacencyMatrix Having Eigenvalue 3. Lin. Alg. Appl. 1, 281 298, 1968. Category Algebraic graph theory Category Matrices numtheory stub sl Seidelova matrika sosednosti ... more details
In algebraic graph theory , the adjacency algebra of a graph mathematics graph G is the algebra ring theory algebra of polynomial s in the adjacencymatrix A G of the graph. It is an example of a matrix algebra object matrix algebra and is the set of the linear combination s of power mathematics power s of A . ref name biggs Algebraic graph theory, by Norman L. Biggs , 1993, ISBN 0521458978, http books.google.com books?id 6TasRmIFOxQC&pg PA9&dq 22adjacency algebra is 22 v onepage&q 22adjacency 20algebra 20is 22&f false p. 9 ref Some other similar mathematical objects are also called adjacency algebra . Properties Properties of the adjacency algebra of G are associated with various spectral graph theory spectral , adjacency and connectivity properties of G . Statement . The number of walk graph theory walk s of length d between vertices i and j is equal to the i , j th element of A sup d sup . ref name biggs Statement . The dimension mathematics dimension of the adjacency algebra of a connected graph of diameter graph theory diameter d is at least d 1. ref name biggs Corollary . A connected graph of diameter d has at least d distinct eigenvalue s. ref name biggs References reflist Category algebraic graph theory combin stub ... more details
to the adjacency list is the adjacencymatrix . For a graph with a sparse matrix sparse adjacencymatrix an adjacency list representation of the graph occupies less space, because it does not use any ... of adjacency lists on a 32 bit computer, an adjacency list for an undirected graph requires about 8 e bytes of storage, where e is the number of edges each edge gives rise to entries in the two adjacency lists and uses four bytes in each. On the other hand, because each entry in an adjacencymatrix .... Then, if 8 e n sup 2 sup 8, the adjacency list representation occupies more space, which is true when d 1 64. Thus a graph must be sparse for an adjacency list representation to be more memory efficient than an adjacencymatrix. However, this analysis is valid only when ... , an adjacencymatrix provides this at once. However, this neighbor test in an adjacency list requires ... , a,c , b,c . In graph theory , an adjacency list is the representation of all Edge geometry edges ... node of the corresponding arc. Typically, adjacency lists are unordered. Application in computer ... this adjacency list representation a adjacent to b,c b adjacent to a,c c adjacent to a,b In computer science , an adjacency list is a data structure for representing graphs. In an adjacency list representation ... to that vertex s adjacency list . For instance, the representation suggested by Guido van Rossum van ... list of the neighbors of each vertex. One difficulty with the adjacency list structure is that it has ... object oriented variant of the adjacency list structure, sometimes called an incidence list , which ... objects in this version of the adjacency list cause it to use more memory than the version in which ... vertex in an adjacency list representation you simply read its adjacency list. With an adjacencymatrix you must instead scan over an entire row, taking Big O notation O n time. If you, instead ... www.SQLSummit.com AdjacencyList.htm Adjacency List Model year 2004 publisher Morgan Kaufmann isbn 1 ... more details
In pragmatics , a branch of linguistics , an adjacency pair is an example of conversational turn taking. An adjacency pair is composed of two Utterance utterances by two speakers, one after the other. The speaking of the first utterance the first pair part, or the first turn provokes a responding utterance the second pair part, or the second turn . For example, a question such as What s your name? requires the addressee to provide an answer in the following turn, thus completing the adjacency pair. A satisfactory response could be I m James . To provide an irrelevant response, or to fail to complete the pair, is noticed as a breach of Gricean maxims conversational maxim . A reply like I m allergic to shellfish would not satisfy the adjacency pair, as it violates Paul Grice Grice s conversational maxim of relevance. Examples of pairs Many actions in conversation are accomplished through established adjacency pairs, examples of which include greeting greeting Heya Oh, hi offer acceptance rejection Would you like to visit the museum with me this evening? I d love to request acceptance rejection Is it OK if I borrow this book? I d rather you didn t, it s due back at the library tomorrow question answer What does this big red button do? It causes two thirds of the universe to implode complaint excuse remedy It s awfully cold in here Oh, sorry, I ll close the window degreeting degreeting See you Yeah, see you later See also Conversational analysis br Pragmatics External links http bank.ugent.be da ap.htm Adjacency pairs with dis preferred seconds http www.sil.org linguistics GlossaryOfLinguisticTerms WhatIsAnAdjacencyPair.htm Short definition of an adjacency pair Category Pragmatics Category Human communication ling stub ... more details
wiktionary TOCright Matrix may refer to Science and mathematics Matrix mathematics , a mathematical object generally represented as an array of numbers Matrix calculus , a notation for calculus operations on matrix spaces Identity matrix Similarity matrix , which scores the similarity between two data points A number of bioinformatic related matrices, including Position specific scoring matrix , which represents a pattern or motif in biological sequences Substitution matrix , which estimates the rate ... PAM matrix , or Point Accepted Mutation matrix, used in scoring sequence alignments BLOSUM BLOcks of Amino Acid SUbstitution Matrix , also used in scoring sequence alignments Matrix biology , with numerous ... Extracellular matrix , any material part of a tissue that is not part of any cell Mitochondrial matrix , the inner part of a mitochondrion, where the Krebs cycle takes place Osteon or bone matrix, a form of connective tissue found in bone Nuclear matrix , an insoluble fraction of the cell nucleus Germinal matrix , an embryonic brain tissue Matrix, another term for a bacterial growth medium Matrix archeology , the soil or sediment surrounding a dig site Harris matrix , an archaeological tool used to describe sediment deposition over time Matrix chemical analysis , the discarded portion of a sample used for chemical analysis Matrix geology , the fine grains between larger grains in igneous or sedimentary rocks, or the rock material in which a fossil, gem etc. is embedded Matrix isolation , a continuous solid phase in which particles precipitates, etc. are embedded Metal matrix composite , a type of composite material Technology Multistate Anti Terrorism Information Exchange MATRIX , a database of US Citizens Matrix numismatics , an intermediate into which the design of a coin is engraved Matrix printing , a mold for shaping the letters Dot matrix , a type of printer or display, usually of text Matrix decoder , a technology which allows a finite number of audio channels to be decoded ... more details
About the 1999 film the franchise it initiated The Matrix franchise other uses Matrix disambiguation Matrix Infobox film name The Matrix image The Matrix Poster.jpg caption Theatrical release poster director ... The Matrix is a 1999 science fiction film science fiction action film written and directed by The Wachowskis ..., and in Australia on Nowrap April 8 , 1999, and is the first installment in the The Matrix franchise Matrix series of films, comic books, video games, and animation. The film depicts a future in which ... activity are used as an energy source. Upon learning this, computer programmer Neo The Matrix ... words. Please do not add excessive detail to this summary. Computer programmer Neo The Matrix Thomas ... alias Neo. He is restless, and driven to learn the meaning of cryptic references to the Matrix appearing on his computer. An infamous female hacker named Trinity The Matrix Trinity Carrie Anne Moss , confirms that a man named Morpheus The Matrix Morpheus Laurence Fishburne , knows the answers he seeks however, three sinister Agent The Matrix Agents , led by Agent Smith Hugo Weaving , arrest Neo ... about the Matrix. Neo swallows the red pill, and he abruptly finds himself in a liquid filled pod ... heat. The world in which Neo grew up was actually the Matrix, a simulated reality of the world ... crew belong to a group of free humans who unplug others from the Matrix and recruit them to their resistance against the machines. They are able to use their understanding of the Matrix s nature to bend ... control over the Matrix. Neo is trained as a member of the rebellion. A socket in Neo s skull, formerly used by the machines to connect him to the Matrix, allows knowledge to be uploaded directly ... training introduces Neo to the key dangers in the Matrix itself. He learns that fatal injuries suffered ... body of anyone still directly connected to the Matrix, whose purpose is to seek out and eliminate ... as the One , the Agents will be no match for him. The group enters the Matrix and takes Neo ... more details
can be represented by the Adjacencymatrixadjacencymatrix M whose row and column indices index ...A logical matrix , binary matrix , relation matrix , Boolean matrix , or 0,1 matrix is a matrix mathematics matrix with entries from the Boolean domain B 0, 1 . Such a matrix can be used to represent a binary relation between a pair of finite set s. Matrix representation of a relation If R is a Binary ... numbers of the matrix, the sets X and Y are indexed with positive integers i ranges from 1 to the cardinality ... , 1, 4 , 2, 2 , 2, 4 , 3, 3 , 4, 4 . The corresponding representation as a Boolean matrix is math begin ... A permutation matrix is a 0,1 matrix, all of whose columns and rows each have exactly one nonzero element. A Costas array is a special case of a permutation matrix An incidence matrix in combinatorics ..., blocks of a block design , or edges of a graph mathematics A design matrix in analysis of variance is a 0,1 matrix with constant row sums. An adjacencymatrix in graph theory is a matrix whose rows and columns represent the vertices and whose entries represent the edges of the graph. The adjacencymatrix of a simple graph simple , undirected graph is a binary symmetric matrix with zero diagonal. The biadjacency matrix of a simple, undirected bipartite graph is a 0,1 matrix, and any 0,1 matrix ... n smooth numbers can be described as a m × n 0,1 matrix, where is the prime counting ... pixel s in only two colors can be represented as a 0,1 matrix in which the 0 s represent pixels of one color and the 1 s represent pixels of the other color. Some properties The matrix representation of the Equality mathematics equality relation on a finite set is an identity matrix , that is, one ... as a semiring , where addition corresponds to logical OR and multiplication to logical AND , the matrix representation of the composition of relations composition of two relations is equal to the matrix product of the matrix representations of these relation. Frequently operations on binary matrices ... more details
In mathematics , computer science and graph theory , a distance matrix is a matrix mathematics matrix two dimensional array containing the distance s, taken pairwise, of a set of points. This matrix will have a size of N × N where N is the number of points, nodes or vertices often in a graph . Comparison with related matrices Comparison with Adjacencymatrix Distance matrices are related to Adjacencymatrixadjacency matrices , with the differences that a the latter only provides the information which vertices are connected but does not tell about costs or distances between the vertices and b an entry of a distance matrix is smaller if two elements are closer, while close connected vertices yield larger entries in an adjacencymatrix. Comparison with Euclidean distance matrix Unlike an Euclidean distance matrix , the matrix does not need to be Symmetric matrix symmetric that is, the values x sub i,j sub do not necessarily equal x sub j,i sub . Similarly, the matrix values are not restricted to non negative Real number reals as they would be in the Euclidean distance matrix but rather can have negative values, zeros or imaginary number s depending on the cost metric and specific use. Although it is often the case, distance matrices are not restricted to being hollow matrix hollow that is, they can have non zero entries on the main diagonal. Examples and uses For example, suppose these data are to be analyzed, where pixel euclidean distance is the Metric mathematics distance metric . Image Clusters.svg frame none Raw data The distance matrix would be class wikitable a b c d e f a 0 184 222 177 216 231 b 184 0 45 123 128 200 c 222 45 0 129 121 203 d 177 123 129 0 46 83 e 216 128 121 46 0 83 f 231 200 203 83 83 0 These data can then be viewed in graphic form ... matrix . See also Data clustering Computer Vision Min plus matrix multiplication DEFAULTSORT Distance Matrix Category Metric geometry Category Bioinformatics Category Matrices mathapplied stub de ... more details
In mathematics , an integer matrix is a matrix mathematics matrix whose entries are all integer s. Examples include binary matrix binary matrices , the zero matrix , the unit matrix , and the adjacencymatrixadjacency matrices used in graph theory , amongst many others. Integer matrices find frequent application in combinatorics . Examples math left begin array cccc 5 & 2 & 6 & 0 4 & 7 & 3 & 8 5 & 9 & 0 & 4 3 & 1 & 0 & 3 9 & 0 & 2 & 1 end array right math and math left begin array ccc 1 & 5 & 0 0 & 9 & 2 1 & 7 & 3 end array right math are both examples of integer matrices. Properties matrix inverse Invertibility of integer matrices is in general more numerically stable than that of non integer matrices. The determinant of an integer matrix is itself an integer, thus the smallest possible magnitude of the determinant of an invertible integer matrix is one , hence where inverses exist they do not become excessively large see condition number . Theorems from matrix theory that infer properties from determinants thus avoid the traps induced by ill conditioned matrix ill conditioned nearly zero determinant real numbers real or floating point valued matrices. The inverse of an integer matrix math M math is again an integer matrix if and only if the determinant of math M math is exactly math 1 math or math 1 math . Integer matrices of determinant math pm 1 math form the group math mathrm SO n, mathbf Z math , which has far reaching applications in arithmetic and geometry. For math n 2 math , it is closely related to the modular group . The intersection of the integer ... polynomial of an integer matrix has integer coefficients. Since the eigenvalue s of a matrix are the roots of the polynomial, the eigenvalues of an integer matrix are algebraic integers . In dimension .... See also Unimodular matrix External links http mathworld.wolfram.com IntegerMatrix.html Integer Matrix at MathWorld Category Matrices Linear algebra stub sl Celo tevil na matrika ... more details
or Fiedler value of G . If we define a signed edge adjacencymatrix M with element M sub ev sub ... sA s 2 D I math where I is the unit matrix, A is the adjacencymatrix, and D is the degree matrix, and s is a complex valued number. Note that normal Laplacian is just math Delta 1 math . As a matrix representation of the negative discrete Laplace operator The Laplacian matrix can be interpreted as a matrix representation of a particular case of the negative discrete Laplace operator . Such an interpretation ...In the mathematics mathematical field of graph theory the Laplacian matrix , sometimes called admittance matrix or Kirchhoff matrix , is a matrix mathematics matrix representation of a graph mathematics ... mathematics spanning tree s for a given graph. The Laplacian matrix can be used to find many other properties ... s inequality from Riemannian Geometry has a discrete analogue involving the Laplacian Matrix, this is perhaps ... Laplacian. Definition Given a simple graph G with n vertices, its Laplacian matrix math L ell i ... Matrix ref math ell i,j begin cases deg v i & mbox if i j 1 & mbox if i neq j mbox and v i mbox is adjacent to v j 0 & mbox otherwise . end cases math That is, it is the difference of the degree matrix and the adjacencymatrix of the graph. In the case of directed graph s, either the degree graph theory indegree or outdegree might be used, depending on the application. The normalized Laplacian matrix ... . end cases math Example Here is a simple example of a labeled graph and its Laplacian matrix. class wikitable Labeled graph Laplacian matrix image 6n graf.svg 175px math left begin array rrrrrr 2 ... & 3 & 0 0 & 0 & 0 & 1 & 0 & 1 end array right math Properties For a graph G and its Laplacian matrix ... matrix positive semidefinite math forall i, lambda i ge 0 quad lambda 0 0 math . The number of times ... connected components in the graph. math lambda 0 math is always 0 because every Laplacian matrix has ... matrix L satisifies math L M text T M ,, math where math M text T math is the transpose matrix ... more details
In mathematics , a hollow matrix may refer to one of several related classes of matrix mathematics matrix . Sparse A hollow matrix may be one with few non zero entries that is, a sparse matrix . ref cite book author Pierre Mass title Optimal Investment Decisions Rules for Action and Criteria for Choice publisher Prentice Hall year 1962 page 142 ref Diagonal entries all zero A hollow matrix may be a square matrix whose Diagonal Matrices diagonal elements are all equal to zero . ref cite book author James E. Gentle title Matrix Algebra Theory, Computations, and Applications in Statistics publisher Springer Verlag year 2007 isbn 0387708723 page 42 ref The most obvious example is the real numbers real skew symmetric matrix skew symmetric matrix. Other examples are the adjacencymatrix of a finite simple graph a distance matrix or Euclidean distance matrix . If A is an n n hollow matrix, then the elements of A are given by math begin array rlll A n times n & & a ij a ij & & 0 & mbox if quad i j, quad 1 le i,j le n. , end array math In other words, any square matrix which takes the form math left begin array ccccc 0 & 0 & & ddots & & & 0 & & & & 0 end array right math is a hollow matrix. For example math left begin array ccccc 0 & 2 & 6 & frac 1 3 & 4 2 & 0 & 4 & 8 & 0 9 & 4 & 0 & 2 & 933 1 & 4 & 4 & 0 & 6 7 & 9 & 23 & 8 & 0 end array right math is an example of a hollow matrix. Properties The trace of A is trivially zero. The linear map represented by A with respect to a fixed basis maps each basis vector e onto the image of the complement of e . Block of zeroes A hollow matrix may be a square n n matrix with an r s block of zeroes where r s n . ref cite book author Paul Cohn authorlink Paul Cohn title Free Ideal Rings and Localization in General Rings publisher Cambridge University Press year 2006 isbn 0 521 85337 0 page 430 ref References reflist DEFAULTSORT Hollow Matrix Category Matrices Linear algebra stub sl Votla matrika ... more details
Unreferenced date December 2009 In the mathematics mathematical field of graph theory the degree matrix is a diagonal matrix which contains information about the degree graph theory degree of each vertex graph theory vertex . It is used together with the adjacencymatrix to construct the Laplacian matrix of a graph. Definition Given a graph math G V,E math with math V n math the degree matrix math D math for math G math is a math n times n math square matrix defined as math d i,j left begin matrix deg v i & mbox if i j 0 & mbox otherwise end matrix right. math Example class wikitable Vertex labeled graph Degree matrix Image 6n graph2.svg 175px math begin pmatrix 4 & 0 & 0 & 0 & 0 & 0 0 & 3 & 0 & 0 & 0 & 0 0 & 0 & 2 & 0 & 0 & 0 0 & 0 & 0 & 3 & 0 & 0 0 & 0 & 0 & 0 & 3 & 0 0 & 0 & 0 & 0 & 0 & 1 end pmatrix math For an undirected graph , the degree graph theory degree of a vertex is the number of edges incident to the vertex. This means that each loop is counted twice. This is because each edge has two endpoints and each endpoint adds to the degree. The degree matrix of a k regular graph has a constant diagonal of math k math DEFAULTSORT Degree Matrix Category Algebraic graph theory Category Matrices eo Grada matrico sl Matrika stopenj vi Ma tr n b c ... more details
In mathematics , an incidence matrix is a matrix mathematics matrix that shows the relationship between two classes of objects. If the first class is X and the second is Y , the matrix has one row for each ... G has two kinds of incidence matrices unoriented and oriented. The incidence matrix or unoriented incidence matrix of G is a p × q matrix math matrix math b ij math , where p and q are the numbers ... the incidence matrix of the undirected graph shown on the right is a matrix consisting of 4 rows corresponding ... pmatrix 1 & 1 & 1 & 0 1 & 0 & 0 & 0 0 & 1 & 0 & 1 0 & 0 & 1 & 1 end pmatrix math The incidence matrix of a directed graph D is a p × q matrix math b ij math where p and q are the number of vertices ... the opposite sign convention. An oriented incidence matrix of an undirected graph G is the incidence matrix, in the sense of directed graphs, of any orientation of G . That is, in the column of edge ... of the oriented incidence matrix, even though that is technically incorrect. The oriented or unoriented incidence matrix of a graph G is related to the adjacencymatrix of its line graph L G by the following theorem math A L G B G T B G 2I q math where math A L G math is the adjacencymatrix of the line graph of G , B G is the incidence matrix, and math I q math is the identity matrix of dimension q. The Kirchhoff matrix is obtained from the oriented incidence matrix M G by the formula math M ... matrix, viewed as a matrix over the integers or real numbers real or complex numbers . The binary cycle space is the null space of its oriented or unoriented incidence matrix, viewed as a matrix over the two element field mathematics field . Signed and bidirected graphs The incidence matrix of a signed graph is a generalization of the oriented incidence matrix. It is the incidence matrix of any ... rows. The line graph and Kirchhoff matrix properties generalize to signed graphs. Multigraph s The definitions of incidence matrix apply to graphs with loop graph theory loops and multiple edges . The column ... more details
network Image Graph for example adjacency matrix.svg 2. Any matrix of the form math ... with polynomial matrix polynomial matrices see Unimodular polynomial matrix . In mathematics , a unimodular matrix M is an integer matrix having determinant 1 or &minus 1. Equivalently, it is an integer matrix that is invertible over the integers there is an integer matrix N which is its ... Unimodular matrices form a subgroup of the general linear group under matrix multiplication , i.e. the following matrices are unimodular Identity matrix The Matrix inverse inverse of a unimodular matrix The Matrix multiplication product of two unimodular matrices Further The Kronecker product of two ... where p and q are the dimensions of A and B , respectively. Concrete examples include Symplectic matrix Symplectic matrices Pascal matrix Pascal matrices Permutation matrix Permutation matrices Total unimodularity A totally unimodular matrix TU matrix is a matrix for which every square invertible matrix non singular submatrix is unimodular. A totally unimodular matrix need not be square itself. From the definition it follows that any totally unimodular matrix has only 0, 1 or &minus 1 entries ... totally unimodular matrices 1. The unoriented incidence matrix of a bipartite graph, which is the coefficient matrix for bipartite matching graph theory matching , is totally unimodular TU . The unoriented incidence matrix of a non bipartite graph is not TU. More generally, in the appendix to a paper ... be an m by n matrix whose rows can be partitioned into two disjoint sets math B math and math C ... , or both in math C math . It was realized later that these conditions define an incidence matrix of a balanced Signed graph Incidence matrix signed graph thus, this example says that the incidence matrix ... for signed graphs without half edges this generalizes the property of the unoriented incidence matrix ... yield a coefficient matrix with these properties and with empty C . Thus, such network flow problems ... more details
Stinson, p.78 ref Given a symmetric conference matrix, the matrix S can be viewed as the Seidel adjacencymatrix of a graph mathematics graph . The graph has n &minus 1 vertices, corresponding to the rows ...In mathematics , a conference matrix also called a C matrix is a square matrix mathematics matrix C with 0 ... matrix I . Thus, if the matrix has order n , C sup T sup C n &minus 1 I . Some ... of conference matrix. Let us normalize C by, first if the more general definition is used , rearranging ... entry is negative. These operations do not change whether a matrix is a conference matrix. Thus, a normalized conference matrix has all 1 s in its first row and column, except for a 0 in the top left corner, and is 0 on the diagonal. Let S be the matrix that remains when the first row and column ... matrix antisymmetric as is the normalized C if its first row is negated , or n is Singly and doubly even oddly even congruent to 2 modulo 4 and S is symmetric matrix symmetric as is the normalized C . Symmetric conference matrices If C is a symmetric conference matrix of order n > ... integers ref Belevitch, p.240 ref there is a clever proof by elementary matrix theory in van Lint .... This graph is strongly regular graph strongly regular of the type called after the matrix a conference ... S to be the Seidel matrix of the Paley graph. The first few possible orders of a symmetric conference matrix are n 2, 6, 10, 14, 18, not 22, since 21 is not a sum of two squares , 26, 30, not 34 since ... that a symmetric conference matrix of that order exists. Order 66 seems to be an open problem. Example The essentially unique conference matrix of order 6 is given by math begin pmatrix ... . Then there is a Paley graph Paley digraph of order q which leads to an antisymmetric conference matrix of order n q 1. The matrix is obtained by taking for S the q × q matrix that has a 1 in position ... conference matrix. This construction solves only a small part of the problem of deciding for which ... more details
graph or Directed graph digraph whose adjacencymatrix is circulant is called a circulant graph ...In linear algebra , a circulant matrix is a special kind of Toeplitz matrix where each row vector is rotated ... on the cyclic group math mathbf Z n mathbf Z . math In cryptography , a circulant matrix is used ... n times n math circulant matrix math C math takes the form math C begin bmatrix c 0 & c n 1 & dots ... n 1 c n 1 & c n 2 & dots & c 1 & c 0 end bmatrix . math A circulant matrix is fully specified by one ... are each cyclic permutation s of the last row. Note that different sources define the circulant matrix ... the first column of the matrix, or with a different direction of shift. Eigenvectors and eigenvalues The eigenvectors of a circulant matrix are given by math v j 1, omega j, omega j 2, ldots, omega j ... of circulant matrix can be computed as math mathrm det C prod j 0 n 1 c 0 c n 1 omega j ... of a matrix, an equivalent formulation is math mathrm det C prod j 0 n 1 c 0 c 1 omega j c 2 omega j ... . math where P is the cyclic permutation matrix given by math P begin bmatrix 0&0& ldots&0&1 1&0& ldots ..., the product math AB math is circulant, and math AB BA math . The eigenvectors of a circulant matrix ... Fourier transform matrix of the same size. The latter matrix is defined by math U n frac 1 sqrt n ... 0 leq j,k n. math Thus, the matrix math U n math diagonalizable matrix diagonalizes C . In fact ... F. author2 link Charles F. Van Loan title Matrix Computations chapter 4.7.7 Circulant Systems publisher Johns Hopkins edition 3rd isbn 978 0 8018 5414 9 year 1996 ref In linear equations Given a matrix equation math mathbf C mathbf x mathbf b , math where math C math is a circulant square matrix ... on the real line or circle. Then, from the perspective of operator theory , a circulant matrix ... are periodic which is the product of the vector of math a i math by the circulant matrix. The discrete Fourier transform then converts convolution into multiplication, which in the matrix setting ... more details
j math , and 0 otherwise this is the adjacencymatrix of links. Divide each row by math k i math where math k i math is the total number of links to other pages from node i . The matrix H is usually ... thumb 240px Fig.1. Google matrix of Wikipedia articles network, written in the bases of PageRank index fragment of top 200 X 200 matrix elements is shown, total size N 3282257 from 19 A Google matrix is a particular stochastic matrix that is used by Google s PageRank algorithm. The matrix represents a graph with edges representing links between pages. The rank of each page can be generated iteratively from the Google matrix using the power method . However, in order for the power method to converge, the matrix must be stochastic, Irreducible mathematics irreducible and aperiodic . H matrix In order to generate the Google matrix, we must first generate a matrix H representing .... G matrix Image Googlematrixcambridge2006.jpg thumb 260px Fig.2. Google matrix of Cambridge University network 2006 , coarse grained matrix elements are written in the bases of PageRank index, total .... We can first generate the stochastic matrix S from H by adding an edge from every sink state math a math to every other node. In the case where there is only one sink state the matrix S is written ... a relation between nodes without a relation with a factor of math alpha math , the matrix will become ... making it aperiodic. The final Google matrix G can be computed as math G alpha S 1 alpha frac 1 N e e T 1 math By the construction the sum of all non negative elements inside each matrix column is equal ... math a math , the Google matrix can be written as math G alpha H alpha a 1 alpha e frac 1 N e T. math Although G is a dense matrix, it is computable using H which is a sparse matrix. Usually for modern directed networks the matrix H has only about ten nonzero elements in a line, thus only about 10 N multiplications are needed to multiply a vector by matrix G 1,2 . An example of the matrix math ... more details
In mathematics , matrix addition is the operation of adding two matrix mathematics matrices by adding the corresponding entries together. However, there are other operations which could also be considered as a kind of addition for matrices, the direct sum and the Kronecker sum . Entrywise sum The usual matrix addition is defined for two matrices of the same dimensions. The sum of two m n pronounced m by n matrices A and B , denoted by A B , is again an m n matrix computed by adding corresponding elements ref Linear Algebra, S. Lipschutz, M. Lipson, Schaum s Outline Series, 2009, ISBN 978 0 07 154352 1 ref ref Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISBN 978 0 521 86153 3 ref math begin align bold A bold B & begin bmatrix a 11 & a 12 & cdots & a 1n a 21 & a 22 & cdots & a 2n vdots & vdots & ddots & vdots a m1 & a m2 & cdots & a mn end bmatrix begin bmatrix b 11 & b 12 & cdots & b 1n b 21 & b 22 & cdots & b 2n ... one matrix from another, as long as they have the same dimensions. A &minus B is computed by subtracting ... pair of matrices A of size m × n and B of size p × q is a matrix of size m p × n q ... matrix , in particular the direct sum of square matrices is a Block matrix Block diagonal matrices block diagonal matrix . The adjacencymatrix of the union of disjoint graph mathematics graphs or multigraph s is the direct sum of their adjacency matrices. Any element in the Direct sum of modules direct ... using the Kronecker product and normal matrix addition. If A is n by n , B is m by m and math mathbf I k math denotes the k by k identity matrix then the Kronecker sum is defined by math mathbf A oplus mathbf B mathbf A otimes mathbf I m mathbf I n otimes mathbf B . math See also Matrix multiplication Referances reflist External links http ncalculators.com matrix 4x4 matrix addition subtraction calculator.htm 4x4 Matrix Addition and Subtraction http planetmath.org encyclopedia DirectSumOfMatrices.html ... more details
thumb right An undirected graph with adjacencymatrix math begin bmatrix 2 & 1 & 0 1 & 0 & 1 0 & 1 & 0 end bmatrix . math The adjacencymatrix of a finite graph is a basic notion of graph theory . ref ...Image Matrix.svg thumb 247px right Specific elements of a matrix are often denoted by a variable with two ... and first column of a matrix A . In mathematics , a matrix plural matrices , or less commonly matrixes is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries . An example of a matrix with six elements is math begin bmatrix 1 & 9 & 13 20 & 55 & 6 end bmatrix . math Matrices of the same size can be matrix addition added or subtracted element by element. The rule for matrix multiplication is more complicated, and two matrices ... in three dimensional space is a linear transformation. If R is a rotation matrix and v is a column vector a matrix with only one column describing the Position vector position of a point in space ... of two matrices is a matrix that represents the function composition composition of two linear ... . If the matrix is Square matrix square , it is possible to deduce some of its properties by computing its determinant . For example, a square matrix has an inverse matrix inverse if and only if its .... Matrix calculus generalizes classical mathematical analysis analytical notions such as derivative ... of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix mathematics Matrix decomposition methods Matrix decomposition ... to the structure of particular matrix structures, e.g. Sparse matrix sparse matrices and diagonal matrix .... Infinite matrices occur in planetary theory and in atomic theory. A simple example is the matrix representing .... Definition A matrix is a rectangle rectangular arrangement of mathematical expressions that can ... . math The horizontal and vertical lines in a matrix are called rows and columns , respectively. The numbers ... 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
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
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
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. mathdab ... more details
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