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Weakly Chained Diagonally Dominant Matrix
In mathematics, the weakly chained diagonally dominant matrices are a family of nonsingular matrices that include the strictly diagonally dominant matrices. Definition Preliminaries We say row i of a complex matrix A = (a_) is strictly diagonally dominant (SDD) if , a_, >\textstyle, a_, . We say A is SDD if all of its rows are SDD. Weakly diagonally dominant (WDD) is defined with \geq instead. The directed graph associated with an m \times m complex matrix A = (a_) is given by the vertices \ and edges defined as follows: there exists an edge from i \rightarrow j if and only if a_ \neq 0. Definition A complex square matrix A is said to be weakly chained diagonally dominant (WCDD) if * A is WDD and * for each row i_1 that is ''not'' SDD, there exists a walk i_1 \rightarrow i_2 \rightarrow \cdots \rightarrow i_k in the directed graph of A ending at an SDD row i_k. Example The m \times m matrix :\begin1\\ -1 & 1\\ & -1 & 1\\ & & \ddots & \ddots\\ & & & -1 & 1 \end is WCD ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
WCDD Venn Diagram
WPZA (107.9 MHz FM, Air1) is a radio station broadcasting a contemporary worship music format. It is licensed to Canton, Illinois and serves the Peoria radio market. The station is currently owned by Educational Media Foundation. History WPZA was known as WBYS-FM for many years on 98.3 FM with an easy listening format. In August 1997, the station was upgraded to 25,000 watts and moved to 107.9 FM. This allowed the station to target the Peoria market, and in late 2004 switched to a Classic Hits format as "CD 107.9". WPZA is one of only 5 radio stations in the Peoria radio market with a Class B signal. In July, 2008, a tornado took down the original tower constructed in 1947 for sister station WBYS, and a new tower was constructed and registered in March, 2009. On December 27, 2010, WCDD changed the format to country, branded as "CD Country 107.9". On December 11, 2020, WCDD was sold to EMF (Educational Media Foundation) for $170,000, pending a flip to Air1. The station ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonsingular Matrices
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix is uniquely determined by , and is called the (multiplicative) ''inverse'' of , denoted by . Matrix inversion is the process of finding the matrix that satisfies the prior equation for a given invertible matrix . A square matrix that is ''not'' invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any finite region on the number line or complex plane, the probability that the matrix is singular is 0, that is, it will "almost never" be singular. Non-square matrices (-by- matrices for which ) do not hav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diagonally Dominant Matrices
In mathematics, a square matrix is said to be diagonally dominant if, for every row of the matrix, the magnitude of the diagonal entry in a row is larger than or equal to the sum of the magnitudes of all the other (non-diagonal) entries in that row. More precisely, the matrix ''A'' is diagonally dominant if :, a_, \geq \sum_ , a_, \quad\text i \, where ''a''''ij'' denotes the entry in the ''i''th row and ''j''th column. Note that this definition uses a weak inequality, and is therefore sometimes called ''weak diagonal dominance''. If a strict inequality (>) is used, this is called ''strict diagonal dominance''. The unqualified term ''diagonal dominance'' can mean both strict and weak diagonal dominance, depending on the context. Variations The definition in the first paragraph sums entries across each row. It is therefore sometimes called ''row diagonal dominance''. If one changes the definition to sum down each column, this is called ''column diagonal dominance''. Any stric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Directed Graph
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pair where * ''V'' is a set whose elements are called '' vertices'', ''nodes'', or ''points''; * ''A'' is a set of ordered pairs of vertices, called ''arcs'', ''directed edges'' (sometimes simply ''edges'' with the corresponding set named ''E'' instead of ''A''), ''arrows'', or ''directed lines''. It differs from an ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of vertices, which are usually called ''edges'', ''links'' or ''lines''. The aforementioned definition does not allow a directed graph to have multiple arrows with the same source and target nodes, but some authors consider a broader definition that allows directed graphs to have such multiple arcs (namely, they allow the arc set to be a m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Walk (graph Theory)
In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges). A directed path (sometimes called dipathGraph Structure Theory: Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Graph Minors, Held June 22 to July 5, 1991p.205/ref>) in a directed graph is a finite or infinite sequence of edges which joins a sequence of distinct vertices, but with the added restriction that the edges be all directed in the same direction. Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts. See e.g. Bondy and Murty (1976), Gibbons (1985), or Diestel (2005). Korte et al. (1990) cover more advanced algorithmic topics concerning paths in graphs. Definitions Walk, trail, and path * A walk is a finite or infinite sequence of edges which joins a sequence of vertices. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Of A WCDD Matrix
Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties * Graph (topology), a topological space resembling a graph in the sense of discrete mathematics * Graph of a function * Graph of a relation * Graph paper * Chart, a means of representing data (also called a graph) Computing * Graph (abstract data type), an abstract data type representing relations or connections * graph (Unix), Unix command-line utility *Conceptual graph, a model for knowledge representation and reasoning Other uses * HMS ''Graph'', a submarine of the UK Royal Navy See also *Complex network *Graf *Graff (other) *Graph database *Grapheme, in linguistics *Graphemics *Graphic (other) *-graphy (suffix from the Greek for "describe," "write" or "draw") *List of information graphics software This is a list of software to create any kind of information graphics: * either includes the abil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irreducibility (mathematics)
In mathematics, the concept of irreducibility is used in several ways. * A polynomial over a field may be an irreducible polynomial if it cannot be factored over that field. * In abstract algebra, irreducible can be an abbreviation for irreducible element of an integral domain; for example an irreducible polynomial. * In representation theory, an irreducible representation is a nontrivial representation with no nontrivial proper subrepresentations. Similarly, an irreducible module is another name for a simple module. * Absolutely irreducible is a term applied to mean irreducible, even after any finite extension of the field of coefficients. It applies in various situations, for example to irreducibility of a linear representation, or of an algebraic variety; where it means just the same as ''irreducible over an algebraic closure''. * In commutative algebra, a commutative ring ''R'' is irreducible if its prime spectrum, that is, the topological space Spec ''R'', is an irreduc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strongly Connected
In the mathematical theory of directed graphs, a graph is said to be strongly connected if every vertex is reachable from every other vertex. The strongly connected components of an arbitrary directed graph form a partition into subgraphs that are themselves strongly connected. It is possible to test the strong connectivity of a graph, or to find its strongly connected components, in linear time (that is, Θ(''V'' + ''E'')). Definitions A directed graph is called strongly connected if there is a path in each direction between each pair of vertices of the graph. That is, a path exists from the first vertex in the pair to the second, and another path exists from the second vertex to the first. In a directed graph ''G'' that may not itself be strongly connected, a pair of vertices ''u'' and ''v'' are said to be strongly connected to each other if there is a path in each direction between them. The binary relation of being strongly connected is an equivalence relation, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
M-matrix
In mathematics, especially linear algebra, an ''M''-matrix is a ''Z''-matrix with eigenvalues whose real parts are nonnegative. The set of non-singular ''M''-matrices are a subset of the class of ''P''-matrices, and also of the class of inverse-positive matrices (i.e. matrices with inverses belonging to the class of positive matrices). The name ''M''-matrix was seemingly originally chosen by Alexander Ostrowski in reference to Hermann Minkowski, who proved that if a Z-matrix has all of its row sums positive, then the determinant of that matrix is positive.. Characterizations An M-matrix is commonly defined as follows: Definition: Let be a real Z-matrix. That is, where for all . Then matrix ''A'' is also an ''M-matrix'' if it can be expressed in the form , where with , for all , where is at least as large as the maximum of the moduli of the eigenvalues of , and is an identity matrix. For the non-singularity of , according to the Perron–Frobenius theorem, it must be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L-matrix
In mathematics, the class of L-matrices are those matrices whose off-diagonal entries are less than or equal to zero and whose diagonal entries are positive; that is, an L-matrix ''L'' satisfies :L=(\ell_);\quad \ell_ > 0; \quad \ell_\leq 0, \quad i\neq j. See also * Z-matrix—every L-matrix is a Z-matrix * Metzler matrix In mathematics, a Metzler matrix is a matrix in which all the off-diagonal components are nonnegative (equal to or greater than zero): : \forall_\, x_ \geq 0. It is named after the American economist Lloyd Metzler. Metzler matrices appear in sta ...—the negation of any L-matrix is a Metzler matrix References Matrices {{matrix-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James H
James is a common English language surname and given name: *James (name), the typically masculine first name James * James (surname), various people with the last name James James or James City may also refer to: People * King James (other), various kings named James * Saint James (other) * James (musician) * James, brother of Jesus Places Canada * James Bay, a large body of water * James, Ontario United Kingdom * James College, a college of the University of York United States * James, Georgia, an unincorporated community * James, Iowa, an unincorporated community * James City, North Carolina * James City County, Virginia ** James City (Virginia Company) ** James City Shire * James City, Pennsylvania * St. James City, Florida Arts, entertainment, and media * ''James'' (2005 film), a Bollywood film * ''James'' (2008 film), an Irish short film * ''James'' (2022 film), an Indian Kannada-language film * James the Red Engine, a character in ''Thomas the Tank En ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relationship With M-matrices
Relationship most often refers to: * Family relations and relatives: consanguinity * Interpersonal relationship, a strong, deep, or close association or acquaintance between two or more people * Correlation and dependence, relationships in mathematics and statistics between two variables or sets of data * Semantic relationship, an ontology component * Romance (love), a connection between two people driven by love and/or sexual attraction Relationship or Relationships may also refer to: Arts and media * "Relationship" (song), by Young Thug featuring Future * "Relationships", an episode of the British TV series ''As Time Goes By'' * The Relationship, an American rock band ** ''The Relationship'' (album), their 2010 album * The Relationships, an English band who played at the 2009 Truck Festival * ''Relationships'', a 1994 album by BeBe & CeCe Winans * ''Relationships'', a 2001 album by Georgie Fame * "Relationship", a song by Lakeside on the 1987 album ''Power'' * "Relatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
