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Matrix theory is a branch of mathematics which focuses on the study of matrices. Initially, a sub-branch of linear algebra, it has grown to cover subjects related to graph theory, algebra, combinatorics and statistics as well. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...
For the square matrix section, see square matrix. ...
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ...
A diagram of a graph with 6 vertices and 7 edges. ...
Algebra is a branch of mathematics which studies structure and quantity. ...
Combinatorics is a odd branch of mathematics that studies collections (usually finite) then constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), with finding largest, smallest, or optimal objects (extremal combinatorics and combinatorial optimization), and with finding algebraic structures these objects may have (algebraic combinatorics). ...
Statistics is a broad mathematical discipline which studies ways to collect, summarize and draw conclusions from data. ...
Overview
A matrix is a rectangular array of numbers. For an elementary article on matrices, their basic properties, and history, see the article matrix (mathematics). In computer programming, an array, also known as a vector or list, is one of the simplest data structures. ...
For the square matrix section, see square matrix. ...
A matrix can be identified with a linear transformation between two vector spaces. Therefore matrix theory is usually considered as a branch of linear algebra. The square matrices play a special role, because the n×n matrices for fixed n have many closure properties. In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...
A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ...
For the square matrix section, see square matrix. ...
In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ...
In graph theory, each labeled graph corresponds to a unique non-negative matrix, the adjacency matrix. A permutation matrix is the matrix representation of a permutation; it is a square matrix with entries 0 and 1, with just one entry 1 in each row and each column. These types of matrices are used in combinatorics. A diagram of a graph with 6 vertices and 7 edges. ...
In mathematics and computer science, the adjacency matrix for a finite graph on n vertices is an n × n matrix in which entry aij is the number of edges from vi to vj in . ...
In linear algebra, a permutation matrix is a binary matrix that has exactly one entry 1 in each row and each column and 0s elsewhere. ...
In mathematics, especially in abstract algebra and related areas, a permutation is a bijection from a finite set X onto itself. ...
Combinatorics is a odd branch of mathematics that studies collections (usually finite) then constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), with finding largest, smallest, or optimal objects (extremal combinatorics and combinatorial optimization), and with finding algebraic structures these objects may have (algebraic combinatorics). ...
The ideas of stochastic matrix and doubly stochastic matrix are important tools to study stochastic processes, in statistics. In mathematics, especially in probability theory and statistics, and also in linear algebra and computer science, a stochastic matrix is a square matrix whose columns are probability vectors, i. ...
In the mathematics of probability, a stochastic process is a random function. ...
Statistics is a broad mathematical discipline which studies ways to collect, summarize and draw conclusions from data. ...
Positive-definite matrices occur in the search for maxima and minima of real-valued functions, when there are several variables. In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ...
A graph illustrating local min/max and global min/max points In mathematics, a point x* is a local maximum of a function f if there exists some ε > 0 such that f(x*) ≥ f(x) for all x with |x-x*| < ε. ...
It is also important to have a theory of matrices over arbitrary rings. In particular, matrices over polynomial rings are used in control theory. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ...
In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ...
In engineering and mathematics, control theory deals with the behavior of dynamical systems over time. ...
Within pure mathematics, matrix rings can provide a rich field of counter examples for mathematical conjectures, amongst other uses.
Some useful theorems In linear algebra, the Cayley-Hamilton theorem (named after the mathematicians Arthur Cayley and William Hamilton) states that every square matrix over the real or complex field, satisfies its own characteristic equation. ...
In linear algebra, the Jordan normal form, also called the Jordan canonical form, named in honor of the 19th and early 20th-century French mathematician Camille Jordan, answers the question, for a given square matrix M over a field K containing the eigenvalues of M, to what extent can M...
In linear algebra, the QR decomposition of a matrix is a decomposition of the matrix into an orthogonal and a triangular matrix. ...
In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation (named after Issai Schur) is an important matrix decomposition. ...
In linear algebra singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with several applications in signal processing and statistics. ...
External links - A Brief History of Linear Algebra and Matrix Theory
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