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''Wronski determinant''
 
''Wronski determinant''
  
The [[Determinant|determinant]] of a system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w0981801.png" /> vector-functions of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w0981802.png" />,
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The [[Determinant|determinant]] of a system of $n$ vector-functions of dimension $n$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w0981803.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$ \phi_i(t) = \{ \phi_i^1(t), \ldots , \phi_i^n(t) \}, \quad i = 1, \ldots, n $$
  
 
of the type
 
of the type

Revision as of 18:50, 29 August 2018

Wronski determinant

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

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

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)


Comments

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].

References

[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 (1889) 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
How to Cite This Entry:
Wronskian. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wronskian&oldid=37039
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article