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. 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 first-order matrix ordinary differential equation is where x ( t ) {displaystyle mathbf {x} (t)} is an n × 1 {displaystyle n imes 1} vector of functions of an underlying variable t {displaystyle t} , x ˙ ( t ) {displaystyle mathbf {dot {x}} (t)} is the vector of first derivatives of these functions, and A ( t ) {displaystyle mathbf {A} (t)} is an n × n {displaystyle n imes n} matrix of coefficients. In the case where A {displaystyle mathbf {A} } 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 A ( t ) {displaystyle mathbf {A} (t)} commutes with its integral ∫ a t A ( s ) d s {displaystyle int _{a}^{t}mathbf {A} (s)ds} then the general solution to the differential equation is where c {displaystyle mathbf {c} } is an n × 1 {displaystyle n imes 1} 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. Below, this solution is displayed in terms of Putzer's algorithm.