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Tensor Reshaping
In multilinear algebra, a reshaping of tensors is any bijection between the set of indices of an order-M tensor and the set of indices of an order-L tensor, where L refers to the set \ of the first positive integers. For each integer m where 1 \le m \le M for a positive integer M, let V_m denote an I_m-dimensional vector space over a field F. Then there are vector space isomorphisms (linear maps) \begin V_1 \otimes \cdots \otimes V_M & \simeq F^ \otimes \cdots \otimes F^ \\ & \simeq F^ \otimes \cdots \otimes F^ \\ & \simeq F^ \otimes F^ \otimes \cdots \otimes F^ \\ & \simeq F^ \otimes F^ \otimes F^ \otimes \cdots \otimes F^ \\ & \,\,\,\vdots \\ & \simeq F^, \end where \pi \in \mathfrak_M is any permutation and \mathfrak_M is the symmetric group on M elements. Via these (and other) vector space isomorphisms, a tensor can be interpreted in several ways as an order-L tensor where L \le M. Coordinate representation The first vector space isomorphism on the list above, V_1 \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Multilinear Algebra
Multilinear algebra is the study of Function (mathematics), functions with multiple vector space, vector-valued Argument of a function, arguments, with the functions being Linear map, linear maps with respect to each argument. It involves concepts such as Matrix (mathematics), matrices, tensors, multivectors, System of linear equations, systems of linear equations, Higher-dimensional space, higher-dimensional spaces, Determinant, determinants, inner product, inner and outer product, outer products, and Dual space, dual spaces. It is a mathematical tool used in engineering, machine learning, physics, and mathematics. Origin While many theoretical concepts and applications involve Vector space, single vectors, mathematicians such as Hermann Grassmann considered structures involving pairs, triplets, and multivectors that generalize Vector (mathematics and physics), vectors. With multiple combinational possibilities, the space of multivectors expands to 2''n'' dimensions, where ''n'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Coordinate Vector
In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimensional Cartesian coordinate system with the basis as the axes of this system. Coordinates are always specified relative to an ordered basis. Bases and their associated coordinate representations let one realize vector spaces and linear transformations concretely as column vectors, row vectors, and matrices; hence, they are useful in calculations. The idea of a coordinate vector can also be used for infinite-dimensional vector spaces, as addressed below. Definition Let ''V'' be a vector space of dimension ''n'' over a field ''F'' and let : B = \ be an ordered basis for ''V''. Then for every v \in V there is a unique linear combination of the basis vectors that equals '' v '': : v = \alpha _1 b_1 + \alpha _2 b_2 + \cdots + \alpha _n b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
GNU Octave
GNU Octave is a scientific programming language for scientific computing and numerical computation. Octave helps in solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly compatible with MATLAB. It may also be used as a Batch processing, batch-oriented language. As part of the GNU Project, it is free software under the terms of the GNU General Public License. History The project was conceived around 1988. At first it was intended to be a companion to a chemical reactor design course. Full development was started by John W. Eaton in 1992. The first alpha release dates back to 4 January 1993 and on 17 February 1994 version 1.0 was released. Version 9.2.0 was released on 7 June 2024. The program is named after Octave Levenspiel, a former professor of the principal author. Levenspiel was known for his ability to perform quick back-of-the-envelope calculations. Development history Developments In addition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Matlab
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages. Although MATLAB is intended primarily for numeric computing, an optional toolbox uses the MuPAD symbolic engine allowing access to symbolic computing abilities. An additional package, Simulink, adds graphical multi-domain simulation and model-based design for dynamic and embedded systems. , MATLAB has more than four million users worldwide. They come from various backgrounds of engineering, science, and economics. , more than 5000 global colleges and universities use MATLAB to support instruction and research. History Origins MATLAB was invented by mathematician and computer programmer Cleve Moler. The idea for MATLAB was base ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Vectorization (mathematics)
In mathematics, especially in linear algebra and matrix theory, the vectorization of a matrix is a linear transformation which converts the matrix into a vector. Specifically, the vectorization of a matrix ''A'', denoted vec(''A''), is the column vector obtained by stacking the columns of the matrix ''A'' on top of one another: \operatorname(A) = _, \ldots, a_, a_, \ldots, a_, \ldots, a_, \ldots, a_\mathrm Here, a_ represents the element in the ''i''-th row and ''j''-th column of ''A'', and the superscript ^\mathrm denotes the transpose. Vectorization expresses, through coordinates, the isomorphism \mathbf^ := \mathbf^m \otimes \mathbf^n \cong \mathbf^ between these (i.e., of matrices and vectors) as vector spaces. For example, for the 2×2 matrix A = \begin a & b \\ c & d \end, the vectorization is \operatorname(A) = \begin a \\ c \\ b \\ d \end. The connection between the vectorization of ''A'' and the vectorization of its transpose is given by the commutation matrix ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called scalar (mathematics), ''scalars''. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars: real numbers and complex numbers. Scalars can also be, more generally, elements of any field (mathematics), field. Vector spaces generalize Euclidean vectors, which allow modeling of Physical quantity, physical quantities (such as forces and velocity) that have not only a Magnitude (mathematics), magnitude, but also a Orientation (geometry), direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix (mathematics), matrices, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Function (mathematics)
In mathematics, a function from a set (mathematics), set to a set assigns to each element of exactly one element of .; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously. The set is called the Domain of a function, domain of the function and the set is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. History of the function concept, Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable function, differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly increased the possible applications of the concept. A f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Tensor Product
In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W denoted . An element of the form v \otimes w is called the tensor product of v and w. An element of V \otimes W is a tensor, and the tensor product of two vectors is sometimes called an ''elementary tensor'' or a ''decomposable tensor''. The elementary tensors span V \otimes W in the sense that every element of V \otimes W is a sum of elementary tensors. If bases are given for V and W, a basis of V \otimes W is formed by all tensor products of a basis element of V and a basis element of W. The tensor product of two vector spaces captures the properties of all bilinear maps in the sense that a bilinear map from V\times W into another vector space Z factors uniquely through a linear map V\otimes W\to Z (see the section below ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Canonical Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is derived . The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may often be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a vector space are isomorphic and cannot be identified. An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a cano ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Standard Basis
In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors, each of whose components are all zero, except one that equals 1. For example, in the case of the Euclidean plane \mathbb^2 formed by the pairs of real numbers, the standard basis is formed by the vectors \mathbf_x = (1,0),\quad \mathbf_y = (0,1). Similarly, the standard basis for the three-dimensional space \mathbb^3 is formed by vectors \mathbf_x = (1,0,0),\quad \mathbf_y = (0,1,0),\quad \mathbf_z=(0,0,1). Here the vector e''x'' points in the ''x'' direction, the vector e''y'' points in the ''y'' direction, and the vector e''z'' points in the ''z'' direction. There are several common notations for standard-basis vectors, including , , , and . These vectors are sometimes written with a hat to emphasize their status as unit vectors (standard unit vectors). These vectors are a basis in the sense that any othe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Basis (linear Algebra)
In mathematics, a Set (mathematics), set of elements of a vector space is called a basis (: bases) if every element of can be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to . The elements of a basis are called . Equivalently, a set is a basis if its elements are linearly independent and every element of is a linear combination of elements of . In other words, a basis is a linearly independent spanning set. A vector space can have several bases; however all the bases have the same number of elements, called the dimension (vector space), dimension of the vector space. This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. Basis vectors find applications in the study of crystal structures and frame of reference, frames of reference. De ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Symmetric Group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \mathrm_n defined over a finite set of n symbols consists of the permutations that can be performed on the n symbols. Since there are n! (n factorial) such permutation operations, the order (number of elements) of the symmetric group \mathrm_n is n!. Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
