Wronskian

Wronski determinant

The determinant of a system of $n$ vector-functions of dimension $n$,

$$\phi_i(t) = \{ \phi_i^1(t), \ldots , \phi_i^n(t) \}, \quad i = 1, \ldots, n \tag{1} \label{eq-1}$$

of the type

The Wronskian of a system of scalar functions

 (2)

which have derivatives up to order (inclusive) is the determinant

 (3)

The concept was first introduced by J. Wronski [1].

If the vector-functions (1) are linearly dependent on a set , then

If the scalar functions (2) are linearly dependent on a set , then

The converse theorems are usually not true: Identical vanishing of a Wronskian on some set is not a sufficient condition for linear dependence of functions on this set.

Let the vector-functions (1) be the solutions of a linear homogeneous -th order system , , with an -dimensional matrix that is continuous on an interval . If these solutions constitute a fundamental system, then

If the Wronskian of these solutions is equal to zero in at least one point of , it is identically equal to zero on , and the functions (1) are linearly dependent. The Liouville formula

where is the trace of the matrix , is applicable.

Let the functions (2) be the solutions of a linear homogeneous -th order equation

with continuous coefficients on the interval . If these solutions constitute a fundamental system, then

If the Wronskian of these solutions is zero in at least one point of , it is identically equal to zero on , and the functions (2) are linearly dependent. The Liouville formula

applies.

References

 [1] J. Hoene-Wronski, "Réfutation de la théorie des fonctions analytiques de Lagrange" , Paris (1812) [2] L.S. Pontryagin, "Ordinary differential equations" , Addison-Wesley (1962) (Translated from Russian)