Let () be an th order differential equation with constant (real) coefficients. To this equation we can associate a linear system of first order differential equations by defining for , where . A solution of this associated system of equations then gives a solution of the original th order equation in the first entry.
Prove that the matrix of coefficients of the system described above is the transpose of the companion matrix of .
Note that for , and . So the corresponding matrix is , where if , if , and 0 otherwise. Then , where if , if , and 0 otherwise. This is precisely the companion matrix of , as desired.