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Pseudo 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 n \times m matrix and y \in \mathcal R(A), the column space of A. If A is nonsingular (which implie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moore–Penrose Inverse
In mathematics, and in particular linear algebra, the Moore–Penrose inverse of a matrix is the most widely known generalization of the inverse matrix. It was independently described by E. H. Moore in 1920, Arne Bjerhammar in 1951, and Roger Penrose in 1955. Earlier, Erik Ivar Fredholm had introduced the concept of a pseudoinverse of integral operators in 1903. When referring to a matrix, the term pseudoinverse, without further specification, is often used to indicate the Moore–Penrose inverse. The term generalized inverse is sometimes used as a synonym for pseudoinverse. A common use of the pseudoinverse is to compute a "best fit" ( least squares) solution to a system of linear equations that lacks a solution (see below under § Applications). Another use is to find the minimum ( Euclidean) norm solution to a system of linear equations with multiple solutions. The pseudoinverse facilitates the statement and proof of results in linear algebra. The pseudoinverse is de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...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^. * 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 ''A''#. The group inverse can be defined, equivalently, by the properties ''AA''#''A'' = ''A'', ''A''#''AA''# = ''A''#, and ''AA''# = ''A''#''A''. * A projection matrix ''P'', defined as a matrix such th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SIAM Journal On Matrix Analysis And Applications
The ''SIAM Journal on Matrix Analysis and Applications'' (until 1989: ''SIAM Journal on Algebraic and Discrete Methods'') 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 is published by the Society for Industrial and Applied Mathematics. The founding editor-in-chief was Gene H. Golub, who established the journal in 1980. The current editor is Michele Benzi ( Scuola Normale Superiore). See also *Michele Benzi Michele Benzi (born 1962 in Bologna) is an Italian mathematician who works as a full professor in the Scuola Normale Superiore in Pisa. He is known for his contributions to numerical linear algebra and its applications, especially to the solu ... External links * Mathematics journals Publications established in 1980 English-language journals Quarte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Mathematical Gazette
''The Mathematical Gazette'' is an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive areas of mathematics." It was established in 1894 by Edward Mann Langley as the successor to the Reports of the Association for the Improvement of Geometrical Teaching. Its publisher is the Mathematical Association. William John Greenstreet was its editor for more than thirty years (1897–1930). Since 2000, the editor is Gerry Leversha. Editors * Edward Mann Langley: 1894-1896 * Francis Sowerby Macaulay: 1896-1897 * William John Greenstreet: 1897-1930 * Alan Broadbent: 1930-1955 * Reuben Goodstein: 1956-1962 * Edwin A. Maxwell Edwin Arthur Maxwell (12 January 1907 – 27 August 1987) was a Scottish mathematician, who worked at Cambridge University for most of his career. Although his contributions to original research were li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also publishes 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. ... [...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 pa ... [...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 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. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ 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 \ m \times m\ complex unitary matrix, \ \mathbf\ 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\ 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^\mathsf\ . The diagonal entries \ \sigma_i = \Sigma_\ of \ \mathbf\ are uniquely determined by and are known as the singular values of . The ... [...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 one or more linear equations involving the same 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 to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by the ordered triple :(x,y,z)=(1,-2,-2), since it makes all three equations valid. The word "system" indicates that the equations are to be considered collectively, rather than individually. In mathematics, the theory of linear systems is the basis and a fundamental part of linear algebra, a subject which is used in most parts of 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 sy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rank Factorization
In mathematics, given a field \mathbb F, nonnegative integers m,n, and a matrix A\in\mathbb F^, a rank decomposition or rank factorization of is a factorization of of the form , where C\in\mathbb F^ and F\in\mathbb F^, where r=\operatorname A is the rank of A. Existence Every finite-dimensional matrix has a rank decomposition: Let A be an m\times n matrix whose column rank is r. Therefore, there are r linearly independent columns in A; equivalently, the dimension of the column space of A is r. Let \mathbf_1, \mathbf_2, \ldots, \mathbf_r be any basis for the column space of A and place them as column vectors to form the m\times r matrix C = \begin\mathbf_1 & \mathbf_2 & \cdots & \mathbf_r\end. Therefore, every column vector of A is a linear combination of the columns of C. To be precise, if A = \begin\mathbf_1 & \mathbf_2 & \cdots & \mathbf_n\end is an m\times n matrix with \mathbf_j as the j-th column, then :\mathbf_j = f_ \mathbf_1 + f_ \mathbf_2 + \cdots + f_ \mathbf_r, whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Matrix
In mathematics, a square matrix is a 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 shearing or rotation. For example, if R is a square matrix representing a rotation (rotation matrix) and \mathbf is a column vector describing the 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_ (''i'' = 1, …, ''n'') 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 the 4×4 matrix above contains the elements , , , . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
