Wronskian

Short description: Determinant of the matrix of first derivatives of a set of functions

In mathematics, the Wronskian of n differentiable functions is the determinant formed with the functions and their n – 1 first derivatives. It was introduced in 1812 by the Polish mathematician Józef Hoene-Wroński, and is used in the study of differential equations, where it can sometimes show the linear independence of a set of solutions.

Definition

The Wronskian of two differentiable functions f  and g is [math]\displaystyle{ W(f,g)=f g' - g f' }[/math].

More generally, for n real- or complex-valued functions f₁, …, f_n, which are n – 1 times differentiable on an interval I, the Wronskian [math]\displaystyle{ W(f_1,\ldots,f_n) }[/math] is a function on [math]\displaystyle{ x\in I }[/math] defined by [math]\displaystyle{ W(f_1, \ldots, f_n) (x)= \det \begin{bmatrix} 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{bmatrix}. }[/math]

This is the determinant of the matrix constructed by placing the functions in the first row, the first derivatives of the functions in the second row, and so on through the [math]\displaystyle{ (n-1)^{\text{th}} }[/math] derivative, thus forming a square matrix.

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

The Wronskian and linear independence

If the functions f_i are linearly dependent, then so are the columns of the Wronskian (since differentiation is a linear operation), and the Wronskian vanishes. Thus, one may show that a set of differentiable functions is linearly independent on an interval by showing that their Wronskian does not vanish identically. It may, however, vanish at isolated points.

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 combine with vanishing of the Wronskian in an interval to imply linear dependence.

• Maxime Bôcher observed that if the functions are analytic, then the vanishing of the Wronskian in an interval implies that they are linearly dependent.
• (Bôcher 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.

Over fields of positive characteristic p the Wronskian may vanish even for linearly independent polynomials; for example, the Wronskian of x^p and 1 is identically 0.

Application to linear differential equations

In general, for an [math]\displaystyle{ n }[/math]th order linear differential equation, if [math]\displaystyle{ (n-1) }[/math] solutions are known, the last one can be determined by using the Wronskian.

Consider the second order differential equation in Lagrange's notation: [math]\displaystyle{ y'' = a(x)y' + b(x)y }[/math] where [math]\displaystyle{ a(x) }[/math], [math]\displaystyle{ b(x) }[/math] are known, and y is the unknown function to be found. Let us call [math]\displaystyle{ y_1, y_2 }[/math] the two solutions of the equation and form their Wronskian [math]\displaystyle{ W(x) = y_1 y'_2 - y_2 y'_1 }[/math]

Then differentiating [math]\displaystyle{ W(x) }[/math] and using the fact that [math]\displaystyle{ y_i }[/math] obey the above differential equation shows that [math]\displaystyle{ W'(x) = a W(x) }[/math]

Therefore, the Wronskian obeys a simple first order differential equation and can be exactly solved: [math]\displaystyle{ W(x) = C~e^{A(x)} }[/math] where [math]\displaystyle{ A'(x)=a(x) }[/math] and [math]\displaystyle{ C }[/math] is a constant.

Now suppose that we know one of the solutions, say [math]\displaystyle{ y_2 }[/math]. Then, by the definition of the Wronskian, [math]\displaystyle{ y_1 }[/math] obeys a first order differential equation: [math]\displaystyle{ y'_1 -\frac{y'_2}{y_2} y_1 = -W(x)/y_2 }[/math] and can be solved exactly (at least in theory).

The method is easily generalized to higher order equations.

Generalized Wronskians

For n functions of several variables, a generalized Wronskian is a 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 single 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).

History

The Wronskian was introduced by Józef Hoene-Wroński (1812) and given its current name by Thomas Muir (1882, Chapter XVIII).

See also

• Variation of parameters
• Moore matrix, analogous to the Wronskian with differentiation replaced by the Frobenius endomorphism over a finite field.
• Alternant matrix
• Vandermonde matrix

References

• Bôcher, Maxime (1900–1901). "The Theory of Linear Dependence". Annals of Mathematics (Princeton University) 2 (1/4): 81–96. doi:10.2307/2007186. ISSN 0003-486X. 
• 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, doi:10.2307/1986214, ISSN 0002-9947, https://www.ams.org/journals/tran/1901-002-02/S0002-9947-1901-1500560-5/S0002-9947-1901-1500560-5.pdf 
• Bostan, Alin; Dumas, Philippe (2010). "Wronskians and Linear Independence". American Mathematical Monthly (Taylor & Francis) 117 (8): 722–727. doi:10.4169/000298910x515785. ISSN 0002-9890. 
• Hartman, Philip (1964), Ordinary Differential Equations, New York: John Wiley & Sons, ISBN 978-0-89871-510-1, https://books.google.com/books?id=CENAPMUEpfoC 
• Hoene-Wroński, Józef (1812), Réfutation de la théorie des fonctions analytiques de Lagrange, Paris 
• Muir, Thomas (1882), A Treatise on the Theorie of Determinants., Macmillan, https://archive.org/details/atreatiseontheo00muirgoog 
• Peano, Giuseppe (1889), "Sur le déterminant wronskien." (in fr), Mathesis IX: 75–76, 110–112, http://www.maa.org/press/periodicals/convergence/peano-on-wronskians-a-translation-appendix-2-translations 
• Hazewinkel, Michiel, ed. (2001), "Wronskian", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page 
• 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 
• 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