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List Of Matrices
This article lists some important classes of matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called ''entries''. Matrices have a long history of both study and application, leading to diverse ways of classifying matrices. A first group is matrices satisfying concrete conditions of the entries, including constant matrices. Important examples include the identity matrix given by : I_n = \begin 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end. and the zero matrix of dimension m \times n. For example: : O_ = \begin 0 & 0 & 0 \\ 0 & 0 & 0 \end. Further ways of classifying matrices are according to their eigenvalues, or by imposing conditions on the product of the matrix with other matrices. Finally, many domains, both in mathematics and other sciences including physics and chemistry, have particular matrices that are applied chiefly in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taxonomy Of Complex Matrices
Taxonomy is the practice and science of categorization or classification. A taxonomy (or taxonomical classification) is a scheme of classification, especially a hierarchical classification, in which things are organized into groups or types. Among other things, a taxonomy can be used to organize and index knowledge (stored as documents, articles, videos, etc.), such as in the form of a library classification system, or a search engine taxonomy, so that users can more easily find the information they are searching for. Many taxonomies are hierarchies (and thus, have an intrinsic tree structure), but not all are. Originally, taxonomy referred only to the categorisation of organisms or a particular categorisation of organisms. In a wider, more general sense, it may refer to a categorisation of things or concepts, as well as to the principles underlying such a categorisation. Taxonomy organizes taxonomic units known as "taxa" (singular "taxon")." Taxonomy is different from mero ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if a_ denotes the entry in the ith row and jth column then for all indices i and j. Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative. In linear algebra, a real symmetric matrix represents a self-adjoint operator represented in an orthonormal basis over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pascal's Triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany, and Italy. The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 at the top (the 0th row). The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the following manner: In row 0 (the topmost row), there is a unique nonzero entry 1. Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0. For example, the initial number of row 1 (or any other row) is 1 (the sum of 0 and 1), whereas the numbers 1 and 3 in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pascal Matrix
In mathematics, particularly matrix theory and combinatorics, a Pascal matrix is a matrix (possibly infinite) containing the binomial coefficients as its elements. It is thus an encoding of Pascal's triangle in matrix form. There are three natural ways to achieve this: as a lower-triangular matrix, an upper-triangular matrix, or a symmetric matrix. For example, the 5 × 5 matrices are: L_5 = \begin 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 \\ 1 & 2 & 1 & 0 & 0 \\ 1 & 3 & 3 & 1 & 0 \\ 1 & 4 & 6 & 4 & 1 \end\,\,\,U_5 = \begin 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & 0 & 1 & 3 & 6 \\ 0 & 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 0 & 1 \end\,\,\,S_5 = \begin 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 & 5 \\ 1 & 3 & 6 & 10 & 15 \\ 1 & 4 & 10 & 20 & 35 \\ 1 & 5 & 15 & 35 & 70 \end=L_5 \times U_5 There are other ways in which Pascal's triangle can be put into matrix form, but these are not easily extended to infinity. Definition The non-zero elements of a Pascal matrix are given by the bi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Of Ones
In mathematics, a matrix of ones or all-ones matrix is a matrix where every entry is equal to one. Examples of standard notation are given below: :J_2 = \begin 1 & 1 \\ 1 & 1 \end;\quad J_3 = \begin 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end;\quad J_ = \begin 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \end;\quad J_ = \begin 1 & 1 \end.\quad Some sources call the all-ones matrix the unit matrix, but that term may also refer to the identity matrix, a different matrix. A vector of ones or all-ones vector is matrix of ones having row or column form; it should not be confused with ''unit vectors''. Properties For an matrix of ones ''J'', the following properties hold: * The trace of ''J'' equals ''n'', and the determinant equals 0 for ''n'' ≥ 2, but equals 1 if ''n'' = 1. * The characteristic polynomial of ''J'' is (x - n)x^. * The minimal polynomial of ''J'' is x^2-nx. * The rank of ''J'' is 1 and the eigenvalues are ''n'' with multiplicity 1 and 0 with multiplicity . * J^k = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Matrix
In mathematics, a nonnegative matrix, written : \mathbf \geq 0, is a matrix in which all the elements are equal to or greater than zero, that is, : x_ \geq 0\qquad \forall . A positive matrix is a matrix in which all the elements are strictly greater than zero. The set of positive matrices is a subset of all non-negative matrices. While such matrices are commonly found, the term is only occasionally used due to the possible confusion with positive-definite matrices, which are different. A matrix which is both non-negative and is positive semidefinite is called a doubly non-negative matrix. A rectangular non-negative matrix can be approximated by a decomposition with two other non-negative matrices via non-negative matrix factorization. Eigenvalues and eigenvectors of square positive matrices are described by the Perron–Frobenius theorem. Properties *The trace and every row and column sum/product of a nonnegative matrix is nonnegative. Inversion The inverse of any non-singul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lehmer Matrix
In mathematics, particularly matrix theory, the ''n×n'' Lehmer matrix (named after Derrick Henry Lehmer) is the constant symmetric matrix defined by :A_ = \begin i/j, & j\ge i \\ j/i, & j''n''. The values of elements diminish toward zero away from the diagonal, where all elements have value 1. The inverse of a Lehmer matrix is a tridiagonal matrix, where the superdiagonal and subdiagonal have strictly negative entries. Consider again the ''n×n'' ''A'' and ''m×m'' ''B'' Lehmer matrices, where ''m''>''n''. A rather peculiar property of their inverses is that ''A−1'' is ''nearly'' a submatrix of ''B−1'', except for the ''A−1n,n'' element, which is not equal to ''B−1n,n''. A Lehmer matrix of order ''n'' has trace ''n''. Examples The 2×2, 3×3 and 4×4 Lehmer matrices and their inverses are shown below. : \begin A_2=\begin 1 & 1/2 \\ 1/2 & 1 \end; & A_2^=\begin 4/3 & -2/3 \\ -2/3 & \end; \\ \\ A_3=\begin 1 & 1/2 & 1/3 \\ 1/2 & 1 & 2/3 \\ 1/3 & 2/ ... [...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] |
Hankel Matrix
In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a square matrix in which each ascending skew-diagonal from left to right is constant, e.g.: \qquad\begin a & b & c & d & e \\ b & c & d & e & f \\ c & d & e & f & g \\ d & e & f & g & h \\ e & f & g & h & i \\ \end. More generally, a Hankel matrix is any n \times n matrix A of the form A = \begin a_ & a_ & a_ & \ldots & \ldots &a_ \\ a_ & a_2 & & & &\vdots \\ a_ & & & & & \vdots \\ \vdots & & & & & a_\\ \vdots & & & & a_& a_ \\ a_ & \ldots & \ldots & a_ & a_ & a_ \end. In terms of the components, if the i,j element of A is denoted with A_, and assuming i\le j, then we have A_ = A_ for all k = 0,...,j-i. Properties * The Hankel matrix is a symmetric matrix. * Let J_n be the n \times n exchange matrix. If H is a m \times n Hankel matrix, then H = T J_n where T is a m \times n Toeplitz matrix. ** If T is real symmetric, then H = T J_n will have the same eigenvalues as T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Matrix
In linear algebra, a Hilbert matrix, introduced by , is a square matrix with entries being the unit fractions : H_ = \frac. For example, this is the 5 × 5 Hilbert matrix: : H = \begin 1 & \frac & \frac & \frac & \frac \\ \frac & \frac & \frac & \frac & \frac \\ \frac & \frac & \frac & \frac & \frac \\ \frac & \frac & \frac & \frac & \frac \\ \frac & \frac & \frac & \frac & \frac \end. The Hilbert matrix can be regarded as derived from the integral : H_ = \int_0^1 x^ \, dx, that is, as a Gramian matrix for powers of ''x''. It arises in the least squares approximation of arbitrary functions by polynomials. The Hilbert matrices are canonical examples of ill-conditioned matrices, being notoriously difficult to use in numerical computation. For example, the 2-norm condition number of the matrix above is about 4.8. Historical note introduced the Hilbert matrix to study the following question in approximation theory: "Assume that , is a real interval. Is it then po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation Matrix
In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say , represents a permutation of elements and, when used to multiply another matrix, say , results in permuting the rows (when pre-multiplying, to form ) or columns (when post-multiplying, to form ) of the matrix . Definition Given a permutation of ''m'' elements, :\pi : \lbrace 1, \ldots, m \rbrace \to \lbrace 1, \ldots, m \rbrace represented in two-line form by :\begin 1 & 2 & \cdots & m \\ \pi(1) & \pi(2) & \cdots & \pi(m) \end, there are two natural ways to associate the permutation with a permutation matrix; namely, starting with the ''m'' × ''m'' identity matrix, , either permute the columns or permute the rows, according to . Both methods of defining permutation matrices appear in the literature and the properties expressed in one representation can be easily converted to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Matrix
A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0, 1) matrix is a matrix with entries from the Boolean domain Such a matrix can be used to represent a binary relation between a pair of finite sets. Matrix representation of a relation If ''R'' is a binary relation between the finite indexed sets ''X'' and ''Y'' (so ), then ''R'' can be represented by the logical matrix ''M'' whose row and column indices index the elements of ''X'' and ''Y'', respectively, such that the entries of ''M'' are defined by :M_ = \begin 1 & (x_i, y_j) \in R, \\ 0 & (x_i, y_j) \not\in R. \end In order to designate the row and column numbers of the matrix, the sets ''X'' and ''Y'' are indexed with positive integers: ''i'' ranges from 1 to the cardinality (size) of ''X'', and ''j'' ranges from 1 to the cardinality of ''Y''. See the entry on indexed sets for more detail. Example The binary relation ''R'' on the set is defined so that ''aRb'' holds if and only if ''a'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
