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
x ˙ ( t ) = A ( t ) x ( t ) {\displaystyle \mathbf {\dot {x}} (t)=\mathbf {A} (t)\mathbf {x} (t)}
where x ( t ) {\displaystyle \mathbf {x} (t)} is an n × 1 {\displaystyle n\times 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\times 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,
x ( t ) = c 1 e λ 1 t u 1 + c 2 e λ 2 t u 2 + ⋯ + c n e λ n t u n , {\displaystyle \mathbf {x} (t)=c_{1}e^{\lambda _{1}t}\mathbf {u} _{1}+c_{2}e^{\lambda _{2}t}\mathbf {u} _{2}+\cdots +c_{n}e^{\lambda _{n}t}\mathbf {u} _{n}~,}
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 Magnus expansion reduces to leading order, and the general solution to the differential equation is
x ( t ) = e ∫ a t A ( s ) d s c , {\displaystyle \mathbf {x} (t)=e^{\int _{a}^{t}\mathbf {A} (s)ds}\mathbf {c} ~,}
where c {\displaystyle \mathbf {c} } is an n × 1 {\displaystyle n\times 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. When this commutation relation is not satisfied, the general solution is provided by the ordered exponential instead .
x ( t ) = OE [ A ] ( t ) c = T { e ∫ 0 t A ( s ) d s } c . {\displaystyle \mathbf {x} (t)=\operatorname {OE} [\mathbf {A} ](t)\mathbf {c} ={\mathcal {T}}\left\{e^{\int _{0}^{t}\mathbf {A} (s)ds}\right\}\mathbf {c} ~.}
Stability and steady state of the matrix system The matrix equation
x ˙ ( t ) = A x ( t ) + b {\displaystyle \mathbf {\dot {x}} (t)=\mathbf {Ax} (t)+\mathbf {b} }
with n×1 parameter constant vector b is stable if and only if all eigenvalues of the constant matrix A have a negative real part. The steady state x* to which it converges if stable is found by setting
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