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 of 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 simple matrix ordinary differential equation is
where is an n×1 vector of functions of an underlying variable, is the vector of first derivatives of these functions, and is an n×n matrix, of which all elements are constants.
Note that by using the Cayley-Hamilton theorem and Vandermonde-type matrices, a solution may be given in a simple form. Below it is developed the solution in terms of Putzer's algorithm.
This differential equation has the following general solution:
where, ... and are the eigenvalues of ;, ... and are the respective eigenvectors of and, .... and are constants.
Read more about Matrix Differential Equation: Stability and Steady State of The Matrix System, Solution in Matrix Form, Putzer Algorithm For Computing, Deconstructed Example of A Matrix Ordinary Differential Equation, Solving Deconstructed Matrix Ordinary Differential Equations, Solved Example of A Matrix ODE, See Also
Famous quotes containing the words matrix, differential and/or equation:
““The matrix is God?”
“In a manner of speaking, although it would be more accurate ... to say that the matrix has a God, since this being’s omniscience and omnipotence are assumed to be limited to the matrix.”
“If it has limits, it isn’t omnipotent.”
“Exactly.... Cyberspace exists, insofar as it can be said to exist, by virtue of human agency.””
—William Gibson (b. 1948)
“But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.”
—Antonin Artaud (1896–1948)
“A nation fights well in proportion to the amount of men and materials it has. And the other equation is that the individual soldier in that army is a more effective soldier the poorer his standard of living has been in the past.”
—Norman Mailer (b. 1923)