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Matrix Difference Equation
A matrix difference equation is a difference equation in which the value of a vector (or sometimes, a matrix) of variables at one point in time is related to its own value at one or more previous points in time, using matrices. The order of the equation is the maximum time gap between any two indicated values of the variable vector. For example, :\mathbf_t = \mathbf\mathbf_ + \mathbf\mathbf_ is an example of a second-order matrix difference equation, in which is an vector of variables and and are matrices. This equation is homogeneous because there is no vector constant term added to the end of the equation. The same equation might also be written as :\mathbf_ = \mathbf\mathbf_ + \mathbf\mathbf_ or as :\mathbf_n = \mathbf\mathbf_ + \mathbf\mathbf_ The most commonly encountered matrix difference equations are first-order. Nonhomogeneous first-order case and the steady state An example of a nonhomogeneous first-order matrix difference equation is :\mathbf_t = \mathb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Difference Equation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riccati Equation
In mathematics, a Riccati equation in the narrowest sense is any first-order ordinary differential equation that is quadratic in the unknown function. In other words, it is an equation of the form : y'(x) = q_0(x) + q_1(x) \, y(x) + q_2(x) \, y^2(x) where q_0(x) \neq 0 and q_2(x) \neq 0. If q_0(x) = 0 the equation reduces to a Bernoulli equation, while if q_2(x) = 0 the equation becomes a first order linear ordinary differential equation. The equation is named after Jacopo Riccati (1676–1754). More generally, the term Riccati equation is used to refer to matrix equations with an analogous quadratic term, which occur in both continuous-time and discrete-time linear-quadratic-Gaussian control. The steady-state (non-dynamic) version of these is referred to as the algebraic Riccati equation. Conversion to a second order linear equation The non-linear Riccati equation can always be converted to a second order linear ordinary differential equation (ODE): If :y'=q_0(x) + ... [...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 comi ... [...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 linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical System
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it. At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Difference Equation
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear relationship of voltage and current in an electrical conductor (Ohm's law), and the relationship of mass and weight. By contrast, more complicated relationships are ''nonlinear''. Generalized for functions in more than one dimension, linearity means the property of a function of being compatible with addition and scaling, also known as the superposition principle. The word linear comes from Latin ''linearis'', "pertaining to or resembling a line". In mathematics In mathematics, a linear map or linear function ''f''(''x'') is a function that satisfies the two properties: * Additivity: . * Homogeneity of degree 1: for all α. These properties are known as the superposition principle. In this definition, ''x'' is not necessarily a re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Difference Equation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Differential Equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. A matrix differential equation contains more than one function stacked into vector form with a matrix relating the functions to their derivatives. For example, a first-order matrix ordinary differential equation is : \mathbf(t) = \mathbf(t)\mathbf(t) where \mathbf(t) is an n \times 1 vector of functions of an underlying variable t, \mathbf(t) is the vector of first derivatives of these functions, and \mathbf(t) is an n \times n matrix of coefficients. In the case where \mathbf is constant and has ''n'' linearly independent eigenvectors, this differential equation has the following general solution, : \mathbf(t) = c_1 e^ \mathbf_1 + c_2 e^ \mathbf_2 + \cdots + c_n e^ \mathbf_n ~, where are the eigenvalues of A; are the respective eigenvectors of A; and are constants. More generally, if \math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Difference Equation
A rational difference equation is a nonlinear difference equation of the form : x_ = \frac~, where the initial conditions x_, x_,\dots, x_ are such that the denominator never vanishes for any . First-order rational difference equation A first-order rational difference equation is a nonlinear difference equation of the form : w_ = \frac. When a,b,c,d and the initial condition w_0 are real numbers, this difference equation is called a Riccati difference equation. Such an equation can be solved by writing w_t as a nonlinear transformation of another variable x_t which itself evolves linearly. Then standard methods can be used to solve the linear difference equation in x_t. Equations of this form arise from the infinite resistor ladder problem. Solving a first-order equation First approach One approach to developing the transformed variable x_t, when ad-bc \neq 0, is to write : y_= \alpha - \frac where \alpha = (a+d)/c and \beta = (ad-bc)/c^ and where w_t = y_t -d/c. Further ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero Matrix
In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero. It also serves as the additive identity of the additive group of m \times n matrices, and is denoted by the symbol O or 0 followed by subscripts corresponding to the dimension of the matrix as the context sees fit. Some examples of zero matrices are : 0_ = \begin 0 \end ,\ 0_ = \begin 0 & 0 \\ 0 & 0 \end ,\ 0_ = \begin 0 & 0 & 0 \\ 0 & 0 & 0 \end .\ Properties The set of m \times n matrices with entries in a ring K forms a ring K_. The zero matrix 0_ \, in K_ \, is the matrix with all entries equal to 0_K \, , where 0_K is the additive identity in K. : 0_ = \begin 0_K & 0_K & \cdots & 0_K \\ 0_K & 0_K & \cdots & 0_K \\ \vdots & \vdots & \ddots & \vdots \\ 0_K & 0_K & \cdots & 0_K \end_ The zero matrix is the additive identity in K_ \, . That is, for all A \in K_ \, it satisfies the equation :0_+A = A + 0_ = A. There is exactly one zero matrix of an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
