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Matrix Equivalence
In linear algebra, two rectangular ''m''-by-''n'' matrices ''A'' and ''B'' are called equivalent if :B = Q^ A P for some invertible ''n''-by-''n'' matrix ''P'' and some invertible ''m''-by-''m'' matrix ''Q''. Equivalent matrices represent the same linear transformation ''V'' → ''W'' under two different choices of a pair of bases of ''V'' and ''W'', with ''P'' and ''Q'' being the change of basis matrices in ''V'' and ''W'' respectively. The notion of equivalence should not be confused with that of similarity, which is only defined for square matrices, and is much more restrictive (similar matrices are certainly equivalent, but equivalent square matrices need not be similar). That notion corresponds to matrices representing the same endomorphism ''V'' → ''V'' under two different choices of a ''single'' basis of ''V'', used both for initial vectors and their images. Properties Matrix equivalence is an equivalence relation on the space of rectangular m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathematics), matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as line (geometry), lines, plane (geometry), planes and rotation (mathematics), rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to Space of functions, function spaces. Linear algebra is also used in most sciences and fields of engineering because it allows mathematical model, modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Rank Of A Matrix
In linear algebra, the rank of a matrix is the dimension of the vector space generated (or spanned) by its columns. p. 48, § 1.16 This corresponds to the maximal number of linearly independent columns of . This, in turn, is identical to the dimension of the vector space spanned by its rows. Rank is thus a measure of the " nondegenerateness" of the system of linear equations and linear transformation encoded by . There are multiple equivalent definitions of rank. A matrix's rank is one of its most fundamental characteristics. The rank is commonly denoted by or ; sometimes the parentheses are not written, as in .Alternative notation includes \rho (\Phi) from and . Main definitions In this section, we give some definitions of the rank of a matrix. Many definitions are possible; see Alternative definitions for several of these. The column rank of is the dimension of the column space of , while the row rank of is the dimension of the row space of . A fundamental result i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Matrix Congruence
Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryotic organism's cells * Matrix (chemical analysis), the non-analyte components of a sample * Matrix (geology), the fine-grained material in which larger objects are embedded * Matrix (composite), the constituent of a composite material * Hair matrix, produces hair * Nail matrix, part of the nail in anatomy Technology * Matrix (mass spectrometry), a compound that promotes the formation of ions * Matrix (numismatics), a tool used in coin manufacturing * Matrix (printing), a mould for casting letters * Matrix (protocol), an open standard for real-time communication * Matrix (record production), or master, a disc used in the production of phonograph records ** Matrix number, of a gramophone record * D ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Row Equivalence
In linear algebra, two matrices are row equivalent if one can be changed to the other by a sequence of elementary row operations. Alternatively, two ''m'' × ''n'' matrices are row equivalent if and only if they have the same row space. The concept is most commonly applied to matrices that represent systems of linear equations, in which case two matrices of the same size are row equivalent if and only if the corresponding homogeneous systems have the same set of solutions, or equivalently the matrices have the same null space. Because elementary row operations are reversible, row equivalence is an equivalence relation. It is commonly denoted by a tilde (~). There is a similar notion of column equivalence, defined by elementary column operations; two matrices are column equivalent if and only if their transpose matrices are row equivalent. Two rectangular matrices that can be converted into one another allowing both elementary row and column operations are ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Principal Ideal Domain
In mathematics, a principal ideal domain, or PID, is an integral domain (that is, a non-zero commutative ring without nonzero zero divisors) in which every ideal is principal (that is, is formed by the multiples of a single element). Some authors such as Bourbaki refer to PIDs as principal rings. Principal ideal domains are mathematical objects that behave like the integers, with respect to divisibility: any element of a PID has a unique factorization into prime elements (so an analogue of the fundamental theorem of arithmetic holds); any two elements of a PID have a greatest common divisor (although it may not be possible to find it using the Euclidean algorithm). If and are elements of a PID without common divisors, then every element of the PID can be written in the form , etc. Principal ideal domains are Noetherian, they are integrally closed, they are unique factorization domains and Dedekind domains. All Euclidean domains and all fields are principal ideal domain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Free Module
In mathematics, a free module is a module that has a ''basis'', that is, a generating set that is linearly independent. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules. Given any set and ring , there is a free -module with basis , which is called the ''free module on'' or ''module of formal'' -''linear combinations'' of the elements of . A free abelian group is precisely a free module over the ring \Z of integers. Definition For a ring R and an R- module M, the set E\subseteq M is a basis for M if: * E is a generating set for M; that is to say, every element of M is a finite sum of elements of E multiplied by coefficients in R; and * E is linearly independent: for every set \\subset E of distinct elements, r_1 e_1 + r_2 e_2 + \cdots + r_n e_n = 0_M implies that r_1 = r_2 = \cdots = r_n = 0_R (where 0_M is the zero element of M and 0_R is the zer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Smith Normal Form
In mathematics, the Smith normal form (sometimes abbreviated SNF) is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID). The Smith normal form of a matrix is diagonal, and can be obtained from the original matrix by multiplying on the left and right by invertible square matrices. In particular, the integers are a PID, so one can always calculate the Smith normal form of an integer matrix. The Smith normal form is very useful for working with finitely generated modules over a PID, and in particular for deducing the structure of a quotient of a free module. It is named after the Irish mathematician Henry John Stephen Smith. Definition Let A be a nonzero m \times n matrix over a principal ideal domain R. There exist invertible m \times m and n \times n-matrices S,T (with entries in R) such that the product SAT is \begin \alpha_1 & 0 & 0 & \cdots & 0 & \cdots & 0 \\ 0 & \alpha_2 & 0 & & & & \\ 0 & 0 & \ddots ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Canonical Form
In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an object and allows it to be identified in a unique way. The distinction between "canonical" and "normal" forms varies from subfield to subfield. In most fields, a canonical form specifies a ''unique'' representation for every object, while a normal form simply specifies its form, without the requirement of uniqueness. The canonical form of a positive integer in decimal representation is a finite sequence of digits that does not begin with zero. More generally, for a class of objects on which an equivalence relation is defined, a canonical form consists in the choice of a specific object in each class. For example: *Jordan normal form is a canonical form for matrix similarity. *The row echelon form is a canonical form, when one considers as equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Elementary Row Operation
In mathematics, an elementary matrix is a square matrix obtained from the application of a single elementary row operation to the identity matrix. The elementary matrices generate the general linear group when is a field. Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while right multiplication (post-multiplication) represents elementary column operations. Elementary row operations are used in Gaussian elimination to reduce a matrix to row echelon form. They are also used in Gauss–Jordan elimination to further reduce the matrix to reduced row echelon form. Elementary row operations There are three types of elementary matrices, which correspond to three types of row operations (respectively, column operations): ;Row switching: A row within the matrix can be switched with another row. : R_i \leftrightarrow R_j ;Row multiplication: Each element in a row can be multiplied by a non-zero constant. It is also known as ''sca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Matrix (mathematics)
In mathematics, a matrix (: matrices) is a rectangle, rectangular array or table of numbers, symbol (formal), symbols, or expression (mathematics), expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two-by-three matrix", a " matrix", or a matrix of dimension . Matrices are commonly used in linear algebra, where they represent linear maps. In geometry, matrices are widely used for specifying and representing geometric transformations (for example rotation (mathematics), rotations) and coordinate changes. In numerical analysis, many computational problems are solved by reducing them to a matrix computation, and this often involves computing with matrices of huge dimensions. Matrices are used in most areas of mathematics and scientific fields, either directly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Equivalence Relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number a is equal to itself (reflexive). If a = b, then b = a (symmetric). If a = b and b = c, then a = c (transitive). Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class. Notation Various notations are used in the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R; the most common are "a \sim b" and "", which are used when R is implicit, and variations of "a \sim_R b", "", or "" to specify R explicitly. Non-equivalence may be written "" or "a \not\equiv b". Definitions A binary relation \,\si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a group is a group homomorphism . In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set ''S'' to itself. In any category, the composition of any two endomorphisms of is again an endomorphism of . It follows that the set of all endomorphisms of forms a monoid, the full transformation monoid, and denoted (or to emphasize the category ). Automorphisms An invertible endomorphism of is called an automorphism. The set of all automorphisms is a subset of with a group structure, called the automorphism group of and denoted . In the following diagram, the arrows denote implication: Endomorphism rings Any two endomorphisms of an abelian group, , can be ad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |