# Difference between revisions of "Wronskian"

Wronski determinant

The determinant of a system of vector-functions of dimension , (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 .

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.

How to Cite This Entry:
Wronskian. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wronskian&oldid=17717
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article