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Bisymmetric Matrix
In mathematics, a bisymmetric matrix is a square matrix that is symmetric about both of its main diagonals. More precisely, an ''n'' × ''n'' matrix ''A'' is bisymmetric if it satisfies both ''A'' = ''AT'' and ''AJ'' = ''JA'' where ''J'' is the ''n'' × ''n'' exchange matrix. For example, any matrix of the form :\begin a & b & c & d & e \\ b & f & g & h & d \\ c & g & i & g & c \\ d & h & g & f & b \\ e & d & c & b & a \end is bisymmetric. Properties *Bisymmetric matrices are both symmetric centrosymmetric and symmetric persymmetric. *The product of two bisymmetric matrices is a centrosymmetric matrix. *Real-valued bisymmetric matrices are precisely those symmetric matrices whose eigenvalues remain the same aside from possible sign changes following pre- or post-multiplication by the exchange matrix. *If ''A'' is a real bisymmetric matrix with distinct eigenvalues, then the matrices that commute Commute, commutation or commutative may refer to: * Co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Symmetry Qtl3
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] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often used to represent simple linear transformations, such as shearing or rotation. For example, if R is a square matrix representing a rotation (rotation matrix) and \mathbf is a column vector describing the position of a point in space, the product R\mathbf yields another column vector describing the position of that point after that rotation. If \mathbf is a row vector, the same transformation can be obtained using where R^ is the transpose of Main diagonal The entries a_ (''i'' = 1, …, ''n'') form the main diagonal of a square matrix. They lie on the imaginary line which runs from the top left corner to the bottom right corner of the matrix. For instance, the main diagonal of the 4×4 matrix above contains the elements , , , . The d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exchange Matrix
In mathematics, especially linear algebra, the exchange matrices (also called the reversal matrix, backward identity, or standard involutory permutation) are special cases of permutation matrix, permutation matrices, where the 1 elements reside on the Main_diagonal#Antidiagonal, antidiagonal and all other elements are zero. In other words, they are 'row-reversed' or 'column-reversed' versions of the identity matrix.. : J_=\begin 0 & 1 \\ 1 & 0 \end;\quad J_ = \begin 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end; \quad J_ = \begin 0 & 0 & \cdots & 0 & 0 & 1 \\ 0 & 0 & \cdots & 0 & 1 & 0 \\ 0 & 0 & \cdots & 1 & 0 & 0 \\ \vdots & \vdots & & \vdots & \vdots & \vdots \\ 0 & 1 & \cdots & 0 & 0 & 0 \\ 1 & 0 & \cdots & 0 & 0 & 0 \end. Definition If ''J'' is an ''n'' × ''n'' exchange matrix, then the elements of ''J'' are J_ = \begin 1, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centrosymmetric Matrix
In mathematics, especially in linear algebra and matrix theory, a centrosymmetric matrix is a matrix which is symmetric about its center. More precisely, an ''n''×''n'' matrix ''A'' = 'A''''i'',''j''is centrosymmetric when its entries satisfy :''A''''i'',''j'' = ''A''''n''−''i'' + 1,''n''−''j'' + 1 for ''i'', ''j'' ∊. If ''J'' denotes the ''n''×''n'' exchange matrix with 1 on the antidiagonal and 0 elsewhere (that is, ''J''''i'',''n'' + 1 − ''i'' = 1; ''J''''i'',''j'' = 0 if ''j'' ≠ ''n'' +1− ''i''), then a matrix ''A'' is centrosymmetric if and only if ''AJ'' = ''JA''. Examples * All 2×2 centrosymmetric matrices have the form \begin a & b \\ b & a \end. * All 3×3 centrosymmetric matrices have the form \begin a & b & c \\ d & e & d \\ c & b & a \end. * Symmetric Toeplitz matrices are centrosymmetric. Algebraic structure and properties *If ''A'' and ''B'' are centrosymmetric matrices over a field ''F'', then so are ''A'' + ''B'' and ''cA'' for any ''c'' in '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Persymmetric Matrix
In mathematics, persymmetric matrix may refer to: # a square matrix which is symmetric with respect to the northeast-to-southwest diagonal; or # a square matrix such that the values on each line perpendicular to the main diagonal are the same for a given line. The first definition is the most common in the recent literature. The designation "Hankel matrix" is often used for matrices satisfying the property in the second definition. Definition 1 Let ''A'' = (''a''''ij'') be an ''n'' × ''n'' matrix. The first definition of ''persymmetric'' requires that :a_ = a_ for all ''i'', ''j''.. See page 193. For example, 5 × 5 persymmetric matrices are of the form :A = \begin a_ & a_ & a_ & a_ & a_ \\ a_ & a_ & a_ & a_ & a_ \\ a_ & a_ & a_ & a_ & a_ \\ a_ & a_ & a_ & a_ & a_ \\ a_ & a_ & a_ & a_ & a_ \end. This can be equivalently expressed as ''AJ'' = ''JA''T where ''J'' is the exchange matrix. A symmetric matrix is a matrix whose values are symmetric in the northwest-to-southeast diag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commuting Matrices
In linear algebra, two matrices A and B are said to commute if AB=BA, or equivalently if their commutator ,B AB-BA is zero. A set of matrices A_1, \ldots, A_k is said to commute if they commute pairwise, meaning that every pair of matrices in the set commute with each other. Characterizations and properties * Commuting matrices preserve each other's eigenspaces. As a consequence, commuting matrices over an algebraically closed field are simultaneously triangularizable; that is, there are bases over which they are both upper triangular. In other words, if A_1,\ldots,A_k commute, there exists a similarity matrix P such that P^ A_i P is upper triangular for all i \in \. The converse is not necessarily true, as the following counterexample shows: *:\begin 1 & 2 \\ 0 & 3 \end\begin 1 & 1 \\ 0 & 1 \end = \begin 1 & 3 \\ 0 & 3 \end \ne \begin 1 & 5 \\ 0 & 3 \end=\begin 1 & 1 \\ 0 & 1 \end\begin 1 & 2 \\ 0 & 3 \end. : However, if the square of the commutator of two matrices is zero, that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Matrix
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix is uniquely determined by , and is called the (multiplicative) ''inverse'' of , denoted by . Matrix inversion is the process of finding the matrix that satisfies the prior equation for a given invertible matrix . A square matrix that is ''not'' invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any finite region on the number line or complex plane, the probability that the matrix is singular is 0, that is, it will "almost never" be singular. Non-square matrices (-by- matrices for which ) do not hav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
