Wronskian

Wronski determinant

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

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

of the type

$$ W ( \phi _{1} (t) \dots \phi _{n} (t)) \quad = \quad \left | \begin{array}{ccc} \phi _{1} ^{1} (t) &\cdots &\phi _{n} ^{1} (t) \\ \vdots &\ddots &\vdots \\ \phi _{1} ^{n} (t) &\cdots &\phi _{n} ^{n} (t) \\ \end{array} \right | . $$

The Wronskian of a system of $ n $ scalar functions

$$ \tag{2} f _{1} (t) \dots f _{n} (t) $$

which have derivatives up to order $ (n - 1 ) $ (inclusive) is the determinant

$$ \tag{3} W (f _{1} (t), \dots, f _{n} (t)) = \left | \begin{array}{ccc} f _{1} (t) &\cdots &f _{n} (t) \\ f _{1} ^ {\ \prime} (t) &\cdots &f _{n} ^ {\ \prime} (t) \\ \vdots &\ddots &\vdots \\ f _{1} ^ {\ (n-1)} (t) &\cdots &f _{n} ^ {\ (n-1)} (t) \\ \end{array} \right | . $$

The concept was first introduced by J. Wronski [1].

If the vector-functions (1) are linearly dependent on a set $ E $, then

$$ W ( \phi _{1} (t), \dots, \phi _{n} (t) ) \quad \equiv \quad 0,\quad\quad t \in E . $$

If the scalar functions (2) are linearly dependent on a set $ E $, then

$$ W (f _{1} (t), \dots, f _{n} (t)) \quad \equiv \quad 0,\quad\quad t \in E . $$

The converse theorems are usually not true: Identical vanishing of a Wronskian on some set is not a sufficient condition for linear dependence of $ n $ functions on this set.

Let the vector-functions (1) be the solutions of a linear homogeneous $ n $-th order system $ x ^ \prime = A(t)x $, $ x \in \mathbf R ^{n} $, with an $ ( n \times n ) $-dimensional matrix $ A(t) $ that is continuous on an interval $ I $. If these solutions constitute a fundamental system, then

$$ W ( \phi _{1} (t), \dots, \phi _{n} (t) ) \quad \neq \quad 0,\quad\quad t \in I. $$

If the Wronskian of these solutions is equal to zero in at least one point of $ I $, it is identically equal to zero on $ I $, and the functions (1) are linearly dependent. The Liouville formula

$$ W ( \phi _{1} (t), \dots, \phi _{n} (t) )\quad = $$

$$ = \quad W ( \phi _{1} ( \tau ), \dots, \phi _{n} ( \tau )) \ \mathop{\rm exp}\nolimits \ \int\limits _ \tau ^ t \mathop{\rm Tr}\nolimits \ A (s) \ ds ,\quad\quad \tau ,\ t \in I, $$

where $ \mathop{\rm Tr}\nolimits \ A(t) $ is the trace of the matrix $ A(t) $, is applicable.

Let the functions (2) be the solutions of a linear homogeneous $ n $-th order equation

$$ y ^{(n)} + p _{1} (t) y ^{(n-1)} + \dots + p _{n-1} (t) y ^ \prime + p _{n} (t) y \quad = \quad 0 $$

with continuous coefficients on the interval $ I $. If these solutions constitute a fundamental system, then

$$ W (f _{1} (t), \dots, f _{n} (t)) \quad \neq \quad 0,\quad\quad t \in I. $$

If the Wronskian of these solutions is zero in at least one point of $ I $, it is identically equal to zero on $ I $, and the functions (2) are linearly dependent. The Liouville formula

$$ W (f _{1} (t), \dots, f _{n} (t))\quad = $$

$$ = \quad W (f _{1} ( \tau ), \dots, f _{n} ( \tau )) \ \mathop{\rm exp}\nolimits \left [ - \int\limits _ \tau ^ t p _{1} (s) \ ds \right ] ,\quad\quad \tau ,\ t \in I, $$

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 $ n $ functions (2) that are not linearly dependent but with vanishing Wronskian was given by G. Peano, [a3].

A sub-Wronskian of order $ i $ for $ \Phi = \{ f _{1}, \dots, f _{n} \} $ is obtained by taking the Wronskian of a subset of size $ i $ of $ \Phi $. Two theorems giving sufficient conditions for linear dependence in terms of Wronskians are as follows. 1) Let $ n> 1 $, $ f _{1}, \dots, f _{n} $ analytic and $ W( \Phi ) \equiv 0 $, then the $ f _{1}, \dots, f _{n} $ are linearly dependent, [a4], [a5]. 2) Let $ n > 1 $, $ W( \Phi ) \equiv 0 $, but at no point of the interval of definition of $ f _{1}, \dots, f _{n} $ do all sub-Wronskians of order $ n - 1 $ vanish simultaneously, then $ \Phi $ 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