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Frobenius Inner Product
In mathematics, the Frobenius inner product is a binary operation that takes two matrices and returns a scalar. It is often denoted \langle \mathbf,\mathbf \rangle_\mathrm. The operation is a component-wise inner product of two matrices as though they are vectors, and satisfies the axioms for an inner product. The two matrices must have the same dimension—same number of rows and columns—but are not restricted to be square matrices. It is named after Ferdinand Georg Frobenius. Definition Given two complex-number-valued ''n''×''m'' matrices A and B, written explicitly as : \mathbf = \,, \quad \mathbf =, the Frobenius inner product is defined as :\langle \mathbf, \mathbf \rangle_\mathrm =\sum_\overline B_ \, = \mathrm\left(\overline \mathbf\right) \equiv \mathrm\left(\mathbf^ \mathbf\right), where the overline denotes the complex conjugate, and \dagger denotes the Hermitian conjugate. Explicitly, this sum is :\begin \langle \mathbf, \mathbf \rangle_\mathrm = & \overl ... [...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] |
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] |
Bilinear Maps
{{disambiguation ...
Bilinear may refer to: * Bilinear sampling (also called "bilinear filtering"), a method in computer graphics for choosing the color of a texture * Bilinear form, a type of mathematical function from a vector space to the underlying field * Bilinear interpolation, an extension of linear interpolation for interpolating functions of two variables on a rectilinear 2D grid * Bilinear map, a type of mathematical function between vector spaces * Bilinear transform, a method of transforming from the S to Z domain in control theory and signal processing * Bilinear transformation (other) Bilinear transformation may refer to: * Bilinear map or bilinear operator * Bilinear transform (signal processing), a type of conformal map used to switch between continuous-time and discrete-time representations * Möbius transformation (complex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Theory
In mathematics, a matrix (: matrices) is a rectangular array or table of numbers, symbols, or 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 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, or through their use in geometry and numerical analysis. '' Square matr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Product Of Hilbert Spaces
In mathematics, and in particular functional analysis, the tensor product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert spaces is another Hilbert space. Roughly speaking, the tensor product is the metric space completion of the ordinary tensor product. This is an example of a topological tensor product. The tensor product allows Hilbert spaces to be collected into a symmetric monoidal category. Definition Since Hilbert spaces have inner products, one would like to introduce an inner product, and thereby a topology, on the tensor product that arises naturally from the inner products on the factors. Let H_1 and H_2 be two Hilbert spaces with inner products \langle\cdot, \cdot\rangle_1 and \langle\cdot, \cdot\rangle_2, respectively. Construct the tensor product of H_1 and H_2 as vector spaces as explained in the article on tensor products. We can turn this vector space tensor product into an inner pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Norm
In the field of mathematics, norms are defined for elements within a vector space. Specifically, when the vector space comprises matrices, such norms are referred to as matrix norms. Matrix norms differ from vector norms in that they must also interact with matrix multiplication. Preliminaries Given a field \ K\ of either real or complex numbers (or any complete subset thereof), let \ K^\ be the -vector space of matrices with m rows and n columns and entries in the field \ K ~. A matrix norm is a norm on \ K^~. Norms are often expressed with double vertical bars (like so: \ \, A\, \ ). Thus, the matrix norm is a function \ \, \cdot\, : K^ \to \R^\ that must satisfy the following properties: For all scalars \ \alpha \in K\ and matrices \ A, B \in K^\ , * \, A\, \ge 0\ (''positive-valued'') * \, A\, = 0 \iff A=0_ (''definite'') * \left\, \alpha\ A \right\, = \left, \alpha \\ \left\, A\right\, \ (''absolutely homogeneous'') * \, A + B \, \le \, A \, + \, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Multiplication
In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the first and the number of columns of the second matrix. The product of matrices and is denoted as . Matrix multiplication was first described by the French mathematician Jacques Philippe Marie Binet in 1812, to represent the composition of functions, composition of linear maps that are represented by matrices. Matrix multiplication is thus a basic tool of linear algebra, and as such has numerous applications in many areas of mathematics, as well as in applied mathematics, statistics, physics, economics, and engineering. Computing matrix products is a central operation in all computational applications of linear algebra. Not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Analysis
In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties. Some particular topics out of many include; operations defined on matrices (such as matrix addition, matrix multiplication and operations derived from these), functions of matrices (such as matrix exponentiation and matrix logarithm, and even sines and cosines etc. of matrices), and the eigenvalues of matrices (eigendecomposition of a matrix, eigenvalue perturbation theory). Matrix spaces The set of all ''m'' × ''n'' matrices over a field ''F'' denoted in this article ''M''''mn''(''F'') form a vector space. Examples of ''F'' include the set of rational numbers \mathbb, the real numbers \mathbb, and set of complex numbers \mathbb. The spaces ''M''''mn''(''F'') and ''M''''pq''(''F'') are different spaces if ''m'' and ''p'' are unequal, and if ''n'' and ''q'' are unequal; for instance ''M''32(''F'') ≠ ''M''23(''F''). Two ''m''&th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kronecker Product
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a specialization of the tensor product (which is denoted by the same symbol) from vectors to matrices and gives the matrix of the tensor product linear map with respect to a standard choice of basis. The Kronecker product is to be distinguished from the usual matrix multiplication, which is an entirely different operation. The Kronecker product is also sometimes called matrix direct product. The Kronecker product is named after the German mathematician Leopold Kronecker (1823–1891), even though there is little evidence that he was the first to define and use it. The Kronecker product has also been called the ''Zehfuss matrix'', and the ''Zehfuss product'', after , who in 1858 described this matrix operation, but Kronecker product is currently the most widely used term. The misattribution to Kronecker rather than Zehfuss wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert–Schmidt Inner Product
In mathematics, Hilbert–Schmidt may refer to * a Hilbert–Schmidt operator In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A \colon H \to H that acts on a Hilbert space H and has finite Hilbert–Schmidt norm \, A\, ^2_ \ \stackrel\ \sum_ \, Ae_i\, ^ ...; ** a Hilbert–Schmidt integral operator; * the Hilbert–Schmidt theorem. {{Disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Norm
In the field of mathematics, norms are defined for elements within a vector space. Specifically, when the vector space comprises matrices, such norms are referred to as matrix norms. Matrix norms differ from vector norms in that they must also interact with matrix multiplication. Preliminaries Given a field \ K\ of either real or complex numbers (or any complete subset thereof), let \ K^\ be the -vector space of matrices with m rows and n columns and entries in the field \ K ~. A matrix norm is a norm on \ K^~. Norms are often expressed with double vertical bars (like so: \ \, A\, \ ). Thus, the matrix norm is a function \ \, \cdot\, : K^ \to \R^\ that must satisfy the following properties: For all scalars \ \alpha \in K\ and matrices \ A, B \in K^\ , * \, A\, \ge 0\ (''positive-valued'') * \, A\, = 0 \iff A=0_ (''definite'') * \left\, \alpha\ A \right\, = \left, \alpha \\ \left\, A\right\, \ (''absolutely homogeneous'') * \, A + B \, \le \, A \, + \, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sesquilinear Form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows one of the arguments to be "twisted" in a semilinear manner, thus the name; which originates from the Latin numerical prefix ''sesqui-'' meaning "one and a half". The basic concept of the dot product – producing a scalar from a pair of vectors – can be generalized by allowing a broader range of scalar values and, perhaps simultaneously, by widening the definition of a vector. A motivating special case is a sesquilinear form on a complex vector space, . This is a map that is linear in one argument and "twists" the linearity of the other argument by complex conjugation (referred to as being antilinear in the other argument). This case arises naturally in mathematical physics applications. Another important case allows the scalars t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
