Wronskian

In mathematics, the Wronskian (or Wrońskian) is a determinant introduced by Józef Hoene-Wroński (1812) and named by Thomas Muir (1882, Chapter XVIII). It is used in the study of differential equations, where it can sometimes show linear independence in a set of solutions.

Definition

The Wronskian of two differentiable functions f  and g is W(f, g) = f g′ – g f ′.

More generally, for n real- or complex-valued functions f₁, . . . , f_n, which are n – 1 times differentiable on an interval I, the Wronskian W(f₁, . . . , f_n) as a function on I is defined by

\( W(f_1, \ldots, f_n) (x)= \begin{vmatrix} f_1(x) & f_2(x) & \cdots & f_n(x) \\ f_1'(x) & f_2'(x) & \cdots & f_n' (x)\\ \vdots & \vdots & \ddots & \vdots \\ f_1^{(n-1)}(x)& f_2^{(n-1)}(x) & \cdots & f_n^{(n-1)}(x) \end{vmatrix},\qquad x\in I. \)

That is, it is the determinant of the matrix constructed by placing the functions in the first row, the first derivative of each function in the second row, and so on through the (n – 1)th derivative, thus forming a square matrix sometimes called a fundamental matrix.

When the functions fi are solutions of a linear differential equation, the Wronskian can be found explicitly using Abel's identity, even if the functions fi are not known explicitly.

The Wronskian and linear independence

If the functions f_i are linearly dependent, then so are the columns of the Wronskian as differentiation is a linear operation, so the Wronskian vanishes. Thus, the Wronskian can be used to show that a set of differentiable functions is linearly independent on an interval by showing that it does not vanish identically.

A common misconception is that W = 0 everywhere implies linear dependence, but Peano (1889) pointed out that the functions x² and |x|x have continuous derivatives and their Wronskian vanishes everywhere, yet they are not linearly dependent in any neighborhood of 0. There are several extra conditions which ensure that the vanishing of the Wronskian in an interval implies linear dependence. Peano (1889) observed that if the functions are analytic, then the vanishing of the Wronskian in an interval implies that they are linearly dependent. Bocher (1901) gave several other conditions for the vanishing of the Wronskian to imply linear dependence; for example, if the Wronskian of n functions is identically zero and the n Wronskians of n – 1 of them do not all vanish at any point then the functions are linearly dependent. Wolsson (1989a) gave a more general condition that together with the vanishing of the Wronskian implies linear dependence.

Generalized Wronskians

For n functions of several variables, a generalized Wronskian is the determinant of an n by n matrix with entries D_i(f_j) (with 0 ≤ i < n), where each D_i is some constant coefficient linear partial differential operator of order i. If the functions are linearly dependent then all generalized Wronskians vanish. As in the 1 variable case the converse is not true in general: if all generalized Wronskians vanish, this does not imply that the functions are linearly dependent. However, the converse is true in many special cases. For example, if the functions are polynomials and all generalized Wronskians vanish, then the functions are linearly dependent. Roth used this result about generalized Wronskians in his proof of Roth's theorem. For more general conditions under which the converse is valid see Wolsson (1989b).

See also

Moore matrix, analogous to the Wronskian with differentiation replaced by the Frobenius endomorphism over a finite field.

Variation of parameters

References

Bôcher, Maxime (1901), "Certain cases in which the vanishing of the Wronskian is a sufficient condition for linear dependence", Transactions of the American Mathematical Society (Providence, R.I.: American Mathematical Society) 2 (2): 139–149, ISSN 0002-9947, JFM 32.0313.02, JSTOR 1986214
Hartman, Philip (1964), Ordinary Differential Equations, New York: John Wiley & Sons, ISBN 978-0-89871-510-1, MR 0171038, Zbl 0125.32102
Hoene-Wronski, J. (1812), Réfutation de la théorie des fonctions analytiques de Lagrange, Paris
Muir, Thomas (1882), A Treatise on the Theorie of Determinants., Macmillan, JFM 15.0118.05
Peano, Giuseppe (1889), "Sur le déterminant wronskien.", Mathesis (in French) IX: 75–76, 110–112, JFM 21.0153.01
Rozov, N. Kh. (2001), "Wronskian", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Wolsson, Kenneth (1989a), "A condition equivalent to linear dependence for functions with vanishing Wronskian", Linear Algebra and its Applications 116: 1–8, doi:10.1016/0024-3795(89)90393-5, ISSN 0024-3795, MR 989712, Zbl 0671.15005
Wolsson, Kenneth (1989b), "Linear dependence of a function set of m variables with vanishing generalized Wronskians", Linear Algebra and its Applications 117: 73–80, doi:10.1016/0024-3795(89)90548-X, ISSN 0024-3795, MR 993032, Zbl 0724.15004