The determinant of a system of vector-functions of dimension ,
of the type
The Wronskian of a system of scalar functions
which have derivatives up to order (inclusive) is the determinant
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
|||J. Hoene-Wronski, "Réfutation de la théorie des fonctions analytiques de Lagrange" , Paris (1812)|
|||L.S. Pontryagin, "Ordinary differential equations" , Addison-Wesley (1962) (Translated from Russian)|
An example of functions (2) that are not linearly dependent but with vanishing Wronskian was given by G. Peano, [a3].
A sub-Wronskian of order for is obtained by taking the Wronskian of a subset of size of . Two theorems giving sufficient conditions for linear dependence in terms of Wronskians are as follows. 1) Let , analytic and , then the are linearly dependent, [a4], [a5]. 2) Let , , but at no point of the interval of definition of do all sub-Wronskians of order vanish simultaneously, then is linearly dependent, [a3].
For more information and results concerning functions of several variables, cf. [a6], [a7].
|[a1]||T.M. Apostol, "Mathematical analysis" , Addison-Wesley (1974)|
|[a2]||P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)|
|[a3]||G. Peano, "Sur le déterminant Wronskian" Mathesis , 9 (1989) pp. 75–76|
|[a4]||M. Böcher, "Certain cases in which the vanishing of the Wronskian is a sufficient condition for linear dependence" Trans. Amer. Math. Soc. , 2 (1901) pp. 139–149|
|[a5]||D.R. Curtis, "The vanishing of the Wronskian and the problem of linear dependence" Math. Ann. , 65 (1908) pp. 282–298|
|[a6]||K. Wolsson, "A condition equivalent to linear dependence for functions with vanishing Wronskian" Linear Alg. Appl. , 116 (1989) pp. 1–8|
|[a7]||K. Wolsson, "Linear dependence of a function set of variables with vanishing generalized Wronskians" Linear Alg. Appl. , 117 (1989) pp. 73–80|
Wronskian. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wronskian&oldid=17717