Editing Matrix exponential (section)

=== Linear differential equation systems ===

{{Main|Matrix differential equation}}

One of the reasons for the importance of the matrix exponential is that it can be used to solve systems of linear [[ordinary differential equations]]. The solution of
<math display="block"> \frac{d}{dt} y(t) = Ay(t), \quad y(0) = y_0, </math>
where {{mvar|A}} is a constant matrix and ''y'' is a column vector, is given by
<math display="block"> y(t) = e^{At} y_0. </math>

The matrix exponential can also be used to solve the inhomogeneous equation
<math display="block"> \frac{d}{dt} y(t) = Ay(t) + z(t), \quad y(0) = y_0. </math>
See the section on [[#Applications|applications]] below for examples.

There is no closed-form solution for differential equations of the form
<math display="block"> \frac{d}{dt} y(t) = A(t) \, y(t), \quad y(0) = y_0, </math>
where {{mvar|A}} is not constant, but the [[Magnus series]] gives the solution as an infinite sum.