Given derivatives and integrals, it is natural to look for relationships connecting them, and to develop a theory of differential equations. Of all these, systems of linear differential equations, and expecially single equations of second order, or pairs of equations of first order, are relatively easy to analyze. They are of especial importance because of the range of applications, and the great frequency with which they occur in all those applications.

- kinds of equations
- the differential calculus of matrices
- linear matrix differential equations
- the matrizant
- uniqueness and periodicity
- the coefficient matrix as a tensor sum