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Matrix Of Ones
In mathematics, a matrix of ones or all-ones matrix is a matrix with every entry equal to one. For example: :J_2 = \begin 1 & 1 \\ 1 & 1 \end,\quad J_3 = \begin 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end,\quad J_ = \begin 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \end,\quad J_ = \begin 1 & 1 \end.\quad Some sources call the all-ones matrix the unit matrix, but that term may also refer to the identity matrix, a different type of matrix. A vector of ones or all-ones vector is matrix of ones having row or column form; it should not be confused with ''unit vectors''. Properties For an matrix of ones ''J'', the following properties hold: * The trace of ''J'' equals ''n'', and the determinant equals 0 for ''n'' ≥ 2, but equals 1 if ''n'' = 1. * The characteristic polynomial of ''J'' is (x - n)x^. * The minimal polynomial of ''J'' is x^2-nx. * The rank of ''J'' is 1 and the eigenvalues are ''n'' with multiplicity 1 and 0 with multiplicity . * J^k = n^ J for k = 1,2,\ldots .. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Idempotent Matrix
In linear algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself. That is, the matrix A is idempotent if and only if A^2 = A. For this product A^2 to be defined, A must necessarily be a square matrix. Viewed this way, idempotent matrices are idempotent elements of matrix rings. Example Examples of 2 \times 2 idempotent matrices are: \begin 1 & 0 \\ 0 & 1 \end \qquad \begin 3 & -6 \\ 1 & -2 \end Examples of 3 \times 3 idempotent matrices are: \begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end \qquad \begin 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end Real 2 × 2 case If a matrix \begina & b \\ c & d \end is idempotent, then * a = a^2 + bc, * b = ab + bd, implying b(1 - a - d) = 0 so b = 0 or d = 1 - a, * c = ca + cd, implying c(1 - a - d) = 0 so c = 0 or d = 1 - a, * d = bc + d^2. Thus, a necessary condition for a 2\times2 matrix to be idempotent is that either it is diagonal or its trace equals 1. For idempotent diagonal matrices, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Groupoid
In abstract algebra, a central groupoid is an algebraic structure defined by a binary operation \cdot on a set of elements that satisfies the equation (a\cdot b)\cdot (b\cdot c)=b. These structures have bijections to the central digraphs, directed graphs that have exactly one two-edge path between every two vertices, and (for finite central groupoids) to the (0,1)-matrices whose squares are the all-ones matrices. As an example, the operation \cdot on points in the Euclidean plane, defined by recombining their Cartesian coordinates as (x_1,y_1)\cdot (x_2,y_2)=(y_1,x_2) is a central groupoid. The same type of recombination defines a central groupoid over the ordered pairs of elements from any set, called a ''natural central groupoid''. As an algebraic structure with a single binary operation, a central groupoid is a special kind of magma or groupoid. Because central groupoids are defined by an equational identity, they form a variety of algebras in which the free objects are c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Matrix
A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0, 1)-matrix is a matrix with entries from the Boolean domain Such a matrix can be used to represent a binary relation between a pair of finite sets. It is an important tool in combinatorial mathematics and theoretical computer science. Matrix representation of a relation If ''R'' is a binary relation between the finite indexed sets ''X'' and ''Y'' (so ), then ''R'' can be represented by the logical matrix ''M'' whose row and column indices index the elements of ''X'' and ''Y'', respectively, such that the entries of ''M'' are defined by :m_ = \begin 1 & (x_i, y_j) \in R, \\ 0 & (x_i, y_j) \not\in R. \end In order to designate the row and column numbers of the matrix, the sets ''X'' and ''Y'' are indexed with positive integers: ''i'' ranges from 1 to the cardinality (size) of ''X'', and ''j'' ranges from 1 to the cardinality of ''Y''. See the article on indexed sets for more detail. The transpose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Root
In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4^2 = (-4)^2 = 16. Every nonnegative real number has a unique nonnegative square root, called the ''principal square root'' or simply ''the square root'' (with a definite article, see below), which is denoted by \sqrt, where the symbol "\sqrt" is called the '' radical sign'' or ''radix''. For example, to express the fact that the principal square root of 9 is 3, we write \sqrt = 3. The term (or number) whose square root is being considered is known as the ''radicand''. The radicand is the number or expression underneath the radical sign, in this case, 9. For non-negative , the principal square root can also be written in exponent notation, as x^. Every positive number has two square roots: \sqrt (which is positive) and -\sqrt (which i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Tree Theorem
In the mathematics, mathematical field of graph theory, Kirchhoff's theorem or Kirchhoff's matrix tree theorem named after Gustav Kirchhoff is a theorem about the number of spanning trees in a graph (discrete mathematics), graph, showing that this number can be computed in polynomial time from the determinant of a submatrix of the graph's Laplacian matrix; specifically, the number is equal to ''any'' Cofactor (linear algebra), cofactor of the Laplacian matrix. Kirchhoff's theorem is a generalization of Cayley's formula which provides the number of spanning trees in a complete graph. Kirchhoff's theorem relies on the notion of the Laplacian matrix of a graph, which is equal to the difference between the graph's degree matrix (the diagonal matrix of vertex degree (graph theory), degrees) and its adjacency matrix (a (0,1)-matrix with 1's at places corresponding to entries where the vertices are adjacent and 0's otherwise). For a given connectivity (graph theory), connected graph ''G' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction). Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete graphs, with their vertices placed on the points of a regular polygon, had already appeared in the 13th century, in the work of Ramon Llull. Such a drawing is sometimes referred to as a mystic rose. Properties The complete graph on vertices is denoted by . Some sources claim that the letter in this notation stands for the German word , but the German name for a complete graph, , does not contain the letter , and other sources state that the notation honors the contributions of Kazimierz Kuratowski to graph theory. has edg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spanning Tree
In the mathematical field of graph theory, a spanning tree ''T'' of an undirected graph ''G'' is a subgraph that is a tree which includes all of the vertices of ''G''. In general, a graph may have several spanning trees, but a graph that is not connected will not contain a spanning tree (see about spanning forests below). If all of the edges of ''G'' are also edges of a spanning tree ''T'' of ''G'', then ''G'' is a tree and is identical to ''T'' (that is, a tree has a unique spanning tree and it is itself). Applications Several pathfinding algorithms, including Dijkstra's algorithm and the A* search algorithm, internally build a spanning tree as an intermediate step in solving the problem. In order to minimize the cost of power networks, wiring connections, piping, automatic speech recognition, etc., people often use algorithms that gradually build a spanning tree (or many such trees) as intermediate steps in the process of finding the minimum spanning tree. The Intern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley's Formula
In mathematics, Cayley's formula is a result in graph theory named after Arthur Cayley. It states that for every positive integer n, the number of trees on n labeled vertices is n^. The formula equivalently counts the spanning trees of a complete graph with labeled vertices . Proof Many proofs of Cayley's tree formula are known. One classical proof of the formula uses Kirchhoff's matrix tree theorem, a formula for the number of spanning trees in an arbitrary graph involving the determinant of a matrix. Prüfer sequences yield a bijective proof of Cayley's formula. Another bijective proof, by André Joyal, finds a one-to-one transformation between ''n''-node trees with two distinguished nodes and maximal directed pseudoforests. A proof by double counting due to Jim Pitman counts in two different ways the number of different sequences of directed edges that can be added to an empty graph on n vertices to form from it a rooted tree; see . History The formula was fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Graph
In graph theory, a regular graph is a Graph (discrete mathematics), graph where each Vertex (graph theory), vertex has the same number of neighbors; i.e. every vertex has the same Degree (graph theory), degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree of each internal vertex are equal to each other. A regular graph with vertices of degree is called a graph or regular graph of degree . Special cases Regular graphs of degree at most 2 are easy to classify: a graph consists of disconnected vertices, a graph consists of disconnected edges, and a graph consists of a disjoint union of graphs, disjoint union of cycle (graph theory), cycles and infinite chains. A graph is known as a cubic graph. A strongly regular graph is a regular graph where every adjacent pair of vertices has the same number of neighbors in common, and every non-adjacent pair of vertices has the same number of neighbors in common. The smal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Undirected Graph
In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called '' vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' means that ''A'' owes money to ''B'', then this graph is directed, because owing money is not necessarily reciprocated. Gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjacency Matrix
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph (discrete mathematics), graph. The elements of the matrix (mathematics), matrix indicate whether pairs of Vertex (graph theory), vertices are Neighbourhood (graph theory), adjacent or not in the graph. In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal. If the graph is Glossary of graph theory terms#undirected, undirected (i.e. all of its Glossary of graph theory terms#edge, edges are bidirectional), the adjacency matrix is symmetric matrix, symmetric. The relationship between a graph and the eigenvalues and eigenvectors of its adjacency matrix is studied in spectral graph theory. The adjacency matrix of a graph should be distinguished from its incidence matrix, a different matrix representation whose elements indicate whether vertex–edge pairs are Incidence (graph), incident or not, and its degree matrix, whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
