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Generalized Inverse
In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element ''x'' is an element ''y'' that has some properties of an inverse element but not necessarily all of them. The purpose of constructing a generalized inverse of a matrix is to obtain a matrix that can serve as an inverse in some sense for a wider class of matrices than invertible matrices. Generalized inverses can be defined in any mathematical structure that involves associative multiplication, that is, in a semigroup. This article describes generalized inverses of a matrix A. A matrix A^\mathrm \in \mathbb^ is a generalized inverse of a matrix A \in \mathbb^ if AA^\mathrmA = A. A generalized inverse exists for an arbitrary matrix, and when a matrix has a regular inverse, this inverse is its unique generalized inverse. Motivation Consider the linear system :Ax = y where A is an m \times n matrix and y \in \mathcal C(A), the column space of A. If m = n and A is nonsingula ... [...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] |
Drazin Inverse
In mathematics, the Drazin inverse, named after Michael P. Drazin, is a kind of generalized inverse of a matrix. Let ''A'' be a square matrix. The index of ''A'' is the least nonnegative integer ''k'' such that rank(''A''''k''+1) = rank(''A''''k''). The Drazin inverse of ''A'' is the unique matrix ''A''D that satisfies :A^A^\text = A^k,\quad A^\textAA^\text = A^\text,\quad AA^\text = A^\textA. It's not a generalized inverse in the classical sense, since A A^\text A \neq A in general. * If ''A'' is invertible with inverse A^, then A^\text = A^. * If ''A'' is a block diagonal matrix :A = \begin B & 0 \\ 0 & N \end where B is invertible with inverse B^ and N is a nilpotent matrix, then :A^D = \begin B^ & 0 \\ 0 & 0 \end * Drazin inversion is invariant under conjugation. If A^\text is the Drazin inverse of A, then P A^\text P^ is the Drazin inverse of PAP^. * The Drazin inverse of a matrix of index 0 or 1 is called the group inverse or -inverse and denoted '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SIAM Journal On Matrix Analysis And Applications
The ''SIAM Journal on Matrix Analysis and Applications'' is a peer-reviewed scientific journal covering matrix analysis and its applications. The relevant applications include signal processing, systems and control theory, statistics, Markov chains, mathematical biology, graph theory, and data science. The journal was originally established as the ''SIAM Journal on Algebraic and Discrete Methods'' in 1980, until it split into '' SIAM Journal on Discrete Mathematics'' and the current title in 1988. The journal is published by the Society for Industrial and Applied Mathematics. The founding editor-in-chief An editor-in-chief (EIC), also known as lead editor or chief editor, is a publication's editorial leader who has final responsibility for its operations and policies. The editor-in-chief heads all departments of the organization and is held accoun ... was Gene H. Golub, who established the journal in 1980. The current editor is Michele Benzi ( Scuola Normale Superiore). Re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Mathematical Gazette
''The Mathematical Gazette'' is a triannual peer-reviewed academic journal published by Cambridge University Press on behalf of the Mathematical Association. It covers mathematics education with a focus on the 15–20 years age range. The journal was established in 1894 by Edward Mann Langley as the successor to the ''Reports of the Association for the Improvement of Geometrical Teaching''. William John Greenstreet was its editor-in-chief for more than thirty years (1897–1930). Since 2000, the editor is Gerry Leversha. Editors-in-chief The following persons are or have been editor-in-chief: Abstracting and indexing The journal is abstracted and indexed in EBSCO databases, Emerging Sources Citation Index, Scopus Scopus is a scientific abstract and citation database, launched by the academic publisher Elsevier as a competitor to older Web of Science in 2004. The ensuing competition between the two databases has been characterized as "intense" and is c ..., and zbMA ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Semigroup
In mathematics, a regular semigroup is a semigroup ''S'' in which every element is regular, i.e., for each element ''a'' in ''S'' there exists an element ''x'' in ''S'' such that . Regular semigroups are one of the most-studied classes of semigroups, and their structure is particularly amenable to study via Green's relations. History Regular semigroups were introduced by J. A. Green in his influential 1951 paper "On the structure of semigroups"; this was also the paper in which Green's relations were introduced. The concept of ''regularity'' in a semigroup was adapted from an analogous condition for rings, already considered by John von Neumann. It was Green's study of regular semigroups which led him to define his celebrated relations. According to a footnote in Green 1951, the suggestion that the notion of regularity be applied to semigroups was first made by David Rees. The term inversive semigroup (French: demi-groupe inversif) was historically used as synonym in the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Block Matrix Pseudoinverse
In mathematics, a block matrix pseudoinverse is a formula for the pseudoinverse of a partitioned matrix. This is useful for decomposing or approximating many algorithms updating parameters in signal processing, which are based on the least squares method. Derivation Consider a column-wise partitioned matrix: : \begin\mathbf A & \mathbf B\end,\quad \mathbf A \in \reals^,\quad \mathbf B \in \reals^,\quad m \geq n + p. If the above matrix is full column rank, the Moore–Penrose inverse matrices of it and its transpose are :\begin \begin\mathbf A & \mathbf B\end^+ &= \left( \begin\mathbf A & \mathbf B\end^\textsf \begin\mathbf A & \mathbf B\end \right)^ \begin\mathbf A & \mathbf B\end^\textsf, \\ \begin \mathbf A^\textsf \\ \mathbf B^\textsf \end^+ &= \begin\mathbf A & \mathbf B\end \left( \begin\mathbf A & \mathbf B\end^\textsf \begin\mathbf A & \mathbf B\end \right)^. \end This computation of the pseudoinverse requires (''n'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular-value Decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix into a rotation, followed by a rescaling followed by another rotation. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any matrix. It is related to the polar decomposition. Specifically, the singular value decomposition of an m \times n complex matrix is a factorization of the form \mathbf = \mathbf, where is an complex unitary matrix, \mathbf \Sigma is an m \times n rectangular diagonal matrix with non-negative real numbers on the diagonal, is an n \times n complex unitary matrix, and \mathbf V^* is the conjugate transpose of . Such decomposition always exists for any complex matrix. If is real, then and can be guaranteed to be real orthogonal matrices; in such contexts, the SVD is often denoted \mathbf U \mathbf \Sigma \mathbf V^\mathrm. The diagonal entries \sigma_i = \Sigma_ of \mathbf \Sigma are uniquely det ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
System Of Linear Equations
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variable (math), variables. For example, : \begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in the three variables . A ''Solution (mathematics), solution'' to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. In the example above, a solution is given by the Tuple, ordered triple (x,y,z)=(1,-2,-2), since it makes all three equations valid. Linear systems are a fundamental part of linear algebra, a subject used in most modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. A Nonlinear system, system of non-linear equations can often be Approximation, approximated by a linear system (see linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rank Factorization
A rank is a position in a hierarchy. It can be formally recognized—for example, cardinal, chief executive officer, general, professor—or unofficial. People Formal ranks * Academic rank * Corporate title * Diplomatic rank * Hierarchy of the Catholic Church * Imperial, royal and noble ranks * Military rank * Police rank Unofficial ranks * Social class * Social position * Social status Either * Seniority Mathematics * Rank (differential topology) * Rank (graph theory) * Rank (linear algebra), the dimension of the vector space generated (or spanned) by a matrix's columns * Rank (set theory) * Rank (type theory) * Rank of an abelian group, the cardinality of a maximal linearly independent subset * Rank of a free module * Rank of a greedoid, the maximal size of a feasible set * Rank of a group, the smallest cardinality of a generating set for the group * Rank of a Lie group – see Cartan subgroup * Rank of a matroid, the maximal size of an independent set * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Matrix
In mathematics, a square matrix is a Matrix (mathematics), matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often used to represent simple linear transformations, such as Shear mapping, shearing or Rotation (mathematics), rotation. For example, if R is a square matrix representing a rotation (rotation matrix) and \mathbf is a column vector describing the Position (vector), position of a point in space, the product R\mathbf yields another column vector describing the position of that point after that rotation. If \mathbf is a row vector, the same transformation can be obtained using where R^ is the transpose of Main diagonal The entries a_ () form the main diagonal of a square matrix. They lie on the imaginary line which runs from the top left corner to the bottom right corner of the matrix. For instance, the main diagonal of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplication
Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''Product (mathematics), product''. Multiplication is often denoted by the cross symbol, , by the mid-line dot operator, , by juxtaposition, or, in programming languages, by an asterisk, . The multiplication of whole numbers may be thought of as repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the ''multiplicand'', as the quantity of the other one, the ''multiplier''; both numbers can be referred to as ''factors''. This is to be distinguished from term (arithmetic), ''terms'', which are added. :a\times b = \underbrace_ . Whether the first factor is the multiplier or the multiplicand may be ambiguous or depend upon context. For example, the expression 3 \times 4 , can be phrased as "3 ti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
