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Tridiagonal Matrix
In linear algebra, a tridiagonal matrix is a band matrix that has nonzero elements only on the main diagonal, the subdiagonal/lower diagonal (the first diagonal below this), and the supradiagonal/upper diagonal (the first diagonal above the main diagonal). For example, the following matrix is tridiagonal: :\begin 1 & 4 & 0 & 0 \\ 3 & 4 & 1 & 0 \\ 0 & 2 & 3 & 4 \\ 0 & 0 & 1 & 3 \\ \end. The determinant of a tridiagonal matrix is given by the ''continuant'' of its elements. An orthogonal transformation of a symmetric (or Hermitian) matrix to tridiagonal form can be done with the Lanczos algorithm. Properties A tridiagonal matrix is a matrix that is both upper and lower Hessenberg matrix. In particular, a tridiagonal matrix is a direct sum of ''p'' 1-by-1 and ''q'' 2-by-2 matrices such that — the dimension of the tridiagonal. Although a general tridiagonal matrix is not necessarily symmetric or Hermitian, many of those that arise when solving linear algebra problems have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tridiagonal Matrix Algorithm
In numerical linear algebra, the tridiagonal matrix algorithm, also known as the Thomas algorithm (named after Llewellyn Thomas), is a simplified form of Gaussian elimination that can be used to solve Tridiagonal matrix, tridiagonal systems of equations. A tridiagonal system for ''n'' unknowns may be written as :a_i x_ + b_i x_i + c_i x_ = d_i, where a_1 = 0 and c_n = 0. : \begin b_1 & c_1 & & & 0 \\ a_2 & b_2 & c_2 & & \\ & a_3 & b_3 & \ddots & \\ & & \ddots & \ddots & c_ \\ 0 & & & a_n & b_n \end \begin x_1 \\ x_2 \\ x_3 \\ \vdots \\ x_n \end = \begin d_1 \\ d_2 \\ d_3 \\ \vdots \\ d_n \end . For such systems, the solution can be obtained in O(n) operations instead of O(n^3) required by Gaussian elimination. A first sweep eliminates the a_i's, and then an (abbreviated) backward substitution produces the solution. Examples of such matrice ... [...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 matrix (mathematics), 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 line (geometry), lines, plane (geometry), planes and rotation (mathematics), rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to Space of functions, function spaces. Linear algebra is also used in most sciences and fields of engineering because it allows mathematical model, 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 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimensional
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on itfor example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on itfor example, both a latitude and longitude are required to locate a point on the surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces. In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found nec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalues And Eigenvectors
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a constant factor \lambda when the linear transformation is applied to it: T\mathbf v=\lambda \mathbf v. The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor \lambda (possibly a negative or complex number). Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. A linear transformation's eigenvectors are those vectors that are only stretched or shrunk, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or shrunk. If the eigenvalue is negative, the eigenvector's direction is reversed. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toeplitz Matrix
In linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant. For instance, the following matrix is a Toeplitz matrix: :\qquad\begin a & b & c & d & e \\ f & a & b & c & d \\ g & f & a & b & c \\ h & g & f & a & b \\ i & h & g & f & a \end. Any n \times n matrix A of the form :A = \begin a_0 & a_ & a_ & \cdots & \cdots & a_ \\ a_1 & a_0 & a_ & \ddots & & \vdots \\ a_2 & a_1 & \ddots & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & a_ & a_ \\ \vdots & & \ddots & a_1 & a_0 & a_ \\ a_ & \cdots & \cdots & a_2 & a_1 & a_0 \end is a Toeplitz matrix. If the i,j element of A is denoted A_ then we have :A_ = A_ = a_. A Toeplitz matrix is not necessarily square. Solving a Toeplitz system A matrix equation of the form :Ax = b is called a Toeplitz system if A is a Toeplitz matrix. If A is an n \times n Toeplitz mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Single-pair Matrix
Single-pair may refer to: * Single-pair high-speed digital subscriber line, a data communications technology * Single-pair shortest-path problem, the problem of finding a path between two vertices such that the sum of the weights of its constituent edges is minimized {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toeplitz Matrices
In linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant. For instance, the following matrix is a Toeplitz matrix: :\qquad\begin a & b & c & d & e \\ f & a & b & c & d \\ g & f & a & b & c \\ h & g & f & a & b \\ i & h & g & f & a \end. Any n \times n matrix A of the form :A = \begin a_0 & a_ & a_ & \cdots & \cdots & a_ \\ a_1 & a_0 & a_ & \ddots & & \vdots \\ a_2 & a_1 & \ddots & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & a_ & a_ \\ \vdots & & \ddots & a_1 & a_0 & a_ \\ a_ & \cdots & \cdots & a_2 & a_1 & a_0 \end is a Toeplitz matrix. If the i,j element of A is denoted A_ then we have :A_ = A_ = a_. A Toeplitz matrix is not necessarily square. Solving a Toeplitz system A matrix equation of the form :Ax = b is called a Toeplitz system if A is a Toeplitz matrix. If A is an n \times n Toeplitz mat ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Matrix
In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by an inverse to undo the operation. An invertible matrix multiplied by its inverse yields the identity matrix. Invertible matrices are the same size as their inverse. Definition An -by- square matrix is called invertible 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 which when multiplied by the original matrix gives the identity matrix. Over a field, a square matrix that is ''not'' invertible is called singular or degener ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuant (mathematics)
In algebra, the continuant is a multivariate polynomial representing the determinant of a tridiagonal matrix and having applications in continued fractions. Definition The ''n''-th ''continuant'' K_n(x_1,\;x_2,\;\ldots,\;x_n) is defined recursively by : K_0 = 1 ; \, : K_1(x_1) = x_1 ; \, : K_n(x_1,\;x_2,\;\ldots,\;x_n) = x_n K_(x_1,\;x_2,\;\ldots,\;x_) + K_(x_1,\;x_2,\;\ldots,\;x_) . \, Properties *The continuant K_n(x_1,\;x_2,\;\ldots,\;x_n) can be computed by taking the sum of all possible products of ''x''1,...,''x''''n'', in which any number of disjoint pairs of consecutive terms are deleted (''Euler's rule''). For example, *: K_5(x_1,\;x_2,\;x_3,\;x_4,\;x_5) = x_1 x_2 x_3 x_4 x_5\; +\; x_3 x_4 x_5\; +\; x_1 x_4 x_5\; +\; x_1 x_2 x_5\; +\; x_1 x_2 x_3\; +\; x_1\; +\; x_3\; +\; x_5. :It follows that continuants are invariant with respect to reversing the order of indeterminates: K_n(x_1,\;\ldots,\;x_n) = K_n(x_n,\;\ldots,\;x_1). *The continuant can be computed as the determ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
