# 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

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

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)

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].