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It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group. Let X be an n × n real or complex matrix. The exponential of X, denoted by e X or exp(X), is the n × n matrix given by the power series = =!
In probability theory, the matrix-exponential distribution is an absolutely continuous distribution with rational Laplace–Stieltjes transform. [1] They were first introduced by David Cox in 1955 as distributions with rational Laplace–Stieltjes transforms .
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time ...
Going the other direction, the matrix exponential of any skew-symmetric matrix is an orthogonal matrix (in fact, special orthogonal). For example, the three-dimensional object physics calls angular velocity is a differential rotation, thus a vector in the Lie algebra s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} tangent to SO(3) .
The Weibull distribution interpolates between the exponential distribution with intensity / when = and a Rayleigh distribution of mode = / when =. The Weibull distribution (usually sufficient in reliability engineering ) is a special case of the three parameter exponentiated Weibull distribution where the additional exponent equals 1.
In mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size.. This is used for defining the exponential of a matrix, which is involved in the closed-form solution of systems of linear differential equations.
In particular, this is the case if the matrix A is independent of t. In the general case, however, the expression above is no longer the solution of the problem. The approach introduced by Magnus to solve the matrix initial-value problem is to express the solution by means of the exponential of a certain n × n matrix function Ω(t, t 0):
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