Global InformationMatrix differential equation information
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
where is an vector of functions of an underlying variable , is the vector of first derivatives of these functions, and is an matrix of coefficients.
In the case where is constant and has n linearly independent eigenvectors, this differential equation has the following general solution,
where λ1, λ2, …, λn are the eigenvalues of A; u1, u2, …, un are the respective eigenvectors of A; and c1, c2, …, cn are constants.
More generally, if commutes with its integral then the Magnus expansion reduces to leading order, and the general solution to the differential equation is
where is an constant vector.
By use of the Cayley–Hamilton theorem and Vandermonde-type matrices, this formal matrix exponential solution may be reduced to a simple form.[1] Below, this solution is displayed in terms of Putzer's algorithm.[2]
- ^ Moya-Cessa, H.; Soto-Eguibar, F. (2011). Differential Equations: An Operational Approach. New Jersey: Rinton Press. ISBN 978-1-58949-060-4.
- ^ Putzer, E. J. (1966). "Avoiding the Jordan Canonical Form in the Discussion of Linear Systems with Constant Coefficients". The American Mathematical Monthly. 73 (1): 2–7. doi:10.1080/00029890.1966.11970714. JSTOR 2313914.