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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 = =!
Cauchy's integral formula from complex analysis can also be used to generalize scalar functions to matrix functions. Cauchy's integral formula states that for any analytic function f defined on a set D ⊂ C, one has = , where C is a closed simple curve inside the domain D enclosing x.
The following identity (Campbell 1897) leads to a special case of the Baker–Campbell–Hausdorff formula. Let G be a matrix Lie group and g its corresponding Lie algebra. Let ad X be the linear operator on g defined by ad X Y = [X,Y] = XY − YX for some fixed X ∈ g. (The adjoint endomorphism encountered above.)
The exponential of a matrix A is defined by =!. Given a matrix B, another matrix A is said to be a matrix logarithm of B if e A = B.. Because the exponential function is not bijective for complex numbers (e.g. = =), numbers can have multiple complex logarithms, and as a consequence of this, some matrices may have more than one logarithm, as explained below.
This formula is an analogue of the classical exponential law + = which holds for all real or complex numbers x and y. If x and y are replaced with matrices A and B, and the exponential replaced with a matrix exponential, it is usually necessary for A and B to commute for the law to
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):
The elementary functions are constructed by composing arithmetic operations, the exponential function (), the natural logarithm (), trigonometric functions (,), and their inverses. The complexity of an elementary function is equivalent to that of its inverse, since all elementary functions are analytic and hence invertible by means of Newton's ...
The formula applies to the case where exp is considered as a map on matrix space over ℝ or C, see matrix exponential. When G = GL( n , C ) or GL( n , R ) , the notions coincide precisely. To compute the differential d exp of exp at X , d exp X : T g X → T G exp( X ) , the standard recipe [ 2 ]
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