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Skew-Hamiltonian Matrix
In linear algebra, skew-Hamiltonian matrices are special matrices which correspond to skew-symmetric bilinear forms on a symplectic vector space. Let ''V'' be a vector space, equipped with a symplectic form \Omega. Such a space must be even-dimensional. A linear map A:\; V \mapsto V is called a skew-Hamiltonian operator with respect to \Omega if the form x, y \mapsto \Omega(A(x), y) is skew-symmetric. Choose a basis e_1, ... e_ in ''V'', such that \Omega is written as \sum_i e_i \wedge e_. Then a linear operator is skew-Hamiltonian with respect to \Omega if and only if its matrix ''A'' satisfies A^T J = J A, where ''J'' is the skew-symmetric matrix :J= \begin 0 & I_n \\ -I_n & 0 \\ \end and ''In'' is the n\times n identity matrix. William C. Waterhouse''The structure of alternating-Hamiltonian matrices'' Linear Algebra and its Applications, Volume 396, 1 February 2005, Pages 385-390 Such matrices are called skew-Hamiltonian. The square of a Hamiltonian matrix is skew-Hamilto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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 lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and fields of engineering, because it allows 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 approximations, using the fact that the differential of a multivariate function at a point is the linear ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a 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 . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra. Therefore, the study of matrices is a large part of linear algebra, and most properties and operations of abstract linear algebra can be expressed in terms of matrices. For example, matrix multiplication represents composition of linear maps. Not all matrices are related to linear algebra. This is, in particular, the case in graph theory, of incidence matrices, and adjacency matrices. ''This article focuses on matrices related to linear algebra, and, unle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bilinear Form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear in each argument separately: * and * and The dot product on \R^n is an example of a bilinear form. The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms. When is the field of complex numbers , one is often more interested in sesquilinear forms, which are similar to bilinear forms but are conjugate linear in one argument. Coordinate representation Let be an -dimensional vector space with basis . The matrix ''A'', defined by is called the ''matrix of the bilinear form'' on the basis . If the matrix represents a vector with respect to this basis, and analogously, represents another vector , then: B(\mathbf, \mathbf) = \mathbf^\textsf A\mathbf = \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Vector Space
In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping that is ; Bilinear: Linear in each argument separately; ; Alternating: holds for all ; and ; Non-degenerate: for all implies that . If the underlying field has characteristic not 2, alternation is equivalent to skew-symmetry. If the characteristic is 2, the skew-symmetry is implied by, but does not imply alternation. In this case every symplectic form is a symmetric form, but not vice versa. Working in a fixed basis, ''ω'' can be represented by a matrix. The conditions above are equivalent to this matrix being skew-symmetric, nonsingular, and hollow (all diagonal entries are zero). This should not be confused with a symplectic matrix, which represents a symplectic transformation of the space. If ''V'' is finite-dimensional, then its dimension must necessarily be even since ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Vector Space
In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping that is ; Bilinear: Linear in each argument separately; ; Alternating: holds for all ; and ; Non-degenerate: for all implies that . If the underlying field has characteristic not 2, alternation is equivalent to skew-symmetry. If the characteristic is 2, the skew-symmetry is implied by, but does not imply alternation. In this case every symplectic form is a symmetric form, but not vice versa. Working in a fixed basis, ''ω'' can be represented by a matrix. The conditions above are equivalent to this matrix being skew-symmetric, nonsingular, and hollow (all diagonal entries are zero). This should not be confused with a symplectic matrix, which represents a symplectic transformation of the space. If ''V'' is finite-dimensional, then its dimension must necessarily be even since ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or can be trivially determined by the context. I_1 = \begin 1 \end ,\ I_2 = \begin 1 & 0 \\ 0 & 1 \end ,\ I_3 = \begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end ,\ \dots ,\ I_n = \begin 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end. The term unit matrix has also been widely used, but the term ''identity matrix'' is now standard. The term ''unit matrix'' is ambiguous, because it is also used for a matrix of ones and for any unit of the ring of all n\times n matrices. In some fields, such as group theory or quantum mechanics, the identity matrix is sometimes denoted by a boldface one, \mathbf, or called "id" (short for identity). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William C
William is a male given name of Germanic origin.Hanks, Hardcastle and Hodges, ''Oxford Dictionary of First Names'', Oxford University Press, 2nd edition, , p. 276. It became very popular in the English language after the Norman conquest of England in 1066,All Things William"Meaning & Origin of the Name"/ref> and remained so throughout the Middle Ages and into the modern era. It is sometimes abbreviated "Wm." Shortened familiar versions in English include Will, Wills, Willy, Willie, Bill, and Billy. A common Irish form is Liam. Scottish diminutives include Wull, Willie or Wullie (as in Oor Wullie or the play ''Douglas''). Female forms are Willa, Willemina, Wilma and Wilhelmina. Etymology William is related to the given name ''Wilhelm'' (cf. Proto-Germanic ᚹᛁᛚᛃᚨᚺᛖᛚᛗᚨᛉ, ''*Wiljahelmaz'' > German ''Wilhelm'' and Old Norse ᚢᛁᛚᛋᛅᚼᛅᛚᛘᛅᛋ, ''Vilhjálmr''). By regular sound changes, the native, inherited English form of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Matrix
In mathematics, a Hamiltonian matrix is a -by- matrix such that is symmetric, where is the skew-symmetric matrix :J = \begin 0_n & I_n \\ -I_n & 0_n \\ \end and is the -by- identity matrix. In other words, is Hamiltonian if and only if where denotes the transpose.. Properties Suppose that the -by- matrix is written as the block matrix : A = \begin a & b \\ c & d \end where , , , and are -by- matrices. Then the condition that be Hamiltonian is equivalent to requiring that the matrices and are symmetric, and that .. Another equivalent condition is that is of the form with symmetric. It follows easily from the definition that the transpose of a Hamiltonian matrix is Hamiltonian. Furthermore, the sum (and any linear combination) of two Hamiltonian matrices is again Hamiltonian, as is their commutator. It follows that the space of all Hamiltonian matrices is a Lie algebra, denoted . The dimension of is . The corresponding Lie group is the symplectic group . This ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heike Fassbender
Heike Fassbender is a German mathematician specializing in numerical linear algebra. She is a professor in the Institute for Computational Mathematics at the Technical University of Braunschweig, and the president for the 2017–2019 term of the Gesellschaft für Angewandte Mathematik und Mechanik (GAMM, the International Association of Applied Mathematics and Mechanics). Education and career Fassbender earned a master's degree in computer science at the University at Buffalo, and completed her Ph.D. at the University of Bremen in 1992. Her dissertation, ''Numerische Verfahren zur diskreten trigonometrischen Polynomapproximation'', was jointly supervised by Angelika Bunse-Gerstner and Ludwig F. Elsner. Fassbender moved to the Technical University of Munich in 2000. She joined the Technical University of Braunschweig faculty in 2002, and was vice president for teaching, studies, and further education at the university from 2008 to 2012. Book Fassbender is the author of the book ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrices
Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchise) * Matrix (mathematics), a rectangular array of numbers, symbols or expressions Matrix (or its plural form matrices) may also refer to: Science and mathematics * Matrix (mathematics), algebraic structure, extension of vector into 2 dimensions * 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 Arts and entertainment Fictional entities * Matrix (comics), two comic book ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
