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$$  
 
$$  
W ( \phi _{1} (t) \dots \phi _{n} (t)) \quad = \quad
+
W ( \phi _{1} (t) \dots \phi _{n} (t)) =  
 
\left |
 
\left |
  
\begin{array}{lll}
+
\begin{array}{ccc}
\phi _{1} ^{1} (t)  &\dots &\phi _{n} ^{1} (t)  \\
+
\phi _{1} ^{1} (t)  &\cdots &\phi _{n} ^{1} (t)  \\
\dots &\dots &\dots \\
+
\vdots &\ddots &\vdots \\
\phi _{1} ^{n} (t)  &\dots &\phi _{n} ^{n} (t)  \\
+
\phi _{1} ^{n} (t)  &\cdots &\phi _{n} ^{n} (t)  \\
 
\end{array}
 
\end{array}
  
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$$ \tag{2}
 
$$ \tag{2}
f _{1} (t) \dots f _{n} (t)
+
f _{1} (t), \dots, f _{n} (t)
 
$$
 
$$
  
  
which have derivatives up to order  $  (n - 1 ) $(
+
which have derivatives up to order  $  (n - 1 ) $ (inclusive) is the determinant
inclusive) is the determinant
 
  
 
$$ \tag{3}
 
$$ \tag{3}
W (f _{1} (t) \dots f _{n} (t)) \quad = \quad
+
W (f _{1} (t), \dots, f _{n} (t)) =  
 
\left |
 
\left |
  
\begin{array}{lll}
+
\begin{array}{ccc}
f _{1} (t)  &\dots &f _{n} (t)  \\
+
f _{1} (t)  &\cdots &f _{n} (t)  \\
f _{1} ^ {\  \prime} (t)  &\dots &f _{n} ^ {\  \prime} (t)  \\
+
f _{1} ^ {\  \prime} (t)  &\cdots &f _{n} ^ {\  \prime} (t)  \\
\dots &\dots &\dots \\
+
\vdots &\ddots &\vdots \\
f _{1} ^ {\  (n-1)} (t)  &\dots &f _{n} ^ {\  (n-1)} (t)  \\
+
f _{1} ^ {\  (n-1)} (t)  &\cdots &f _{n} ^ {\  (n-1)} (t)  \\
 
\end{array}
 
\end{array}
  
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$$  
 
$$  
W ( \phi _{1} (t) \dots \phi _{n} (t) ) \quad \equiv \quad 0,\quad\quad
+
W ( \phi _{1} (t), \dots, \phi _{n} (t) ) \quad \equiv \quad 0,\quad\quad
 
t \in E .
 
t \in E .
 
$$
 
$$
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$$  
 
$$  
W (f _{1} (t) \dots f _{n} (t)) \quad \equiv \quad 0,\quad\quad
+
W (f _{1} (t), \dots, f _{n} (t)) \quad \equiv \quad 0,\quad\quad
 
t \in E .
 
t \in E .
 
$$
 
$$
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functions on this set.
 
functions on this set.
  
Let the vector-functions (1) be the solutions of a linear homogeneous  $  n $-
+
Let the vector-functions (1) be the solutions of a linear homogeneous  $  n $-th order system  $  x ^ \prime  = A(t)x $,  
th order system  $  x ^ \prime  = A(t)x $,  
 
 
$  x \in \mathbf R ^{n} $,  
 
$  x \in \mathbf R ^{n} $,  
with an  $  ( n \times n ) $-
+
with an  $  ( n \times n ) $-dimensional matrix  $  A(t) $
dimensional matrix  $  A(t) $
 
 
that is continuous on an interval  $  I $.  
 
that is continuous on an interval  $  I $.  
 
If these solutions constitute a fundamental system, then
 
If these solutions constitute a fundamental system, then
  
 
$$  
 
$$  
W ( \phi _{1} (t) \dots \phi _{n} (t) ) \quad \neq \quad 0,\quad\quad
+
W ( \phi _{1} (t), \dots, \phi _{n} (t) ) \quad \neq \quad 0,\quad\quad
 
t \in I.
 
t \in I.
 
$$
 
$$
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$$  
 
$$  
W ( \phi _{1} (t) \dots \phi _{n} (t) )\quad =
+
W ( \phi _{1} (t), \dots, \phi _{n} (t) )\quad =
 
$$
 
$$
  
Line 97: Line 94:
 
$$  
 
$$  
 
= \quad
 
= \quad
W ( \phi _{1} ( \tau ) \dots \phi _{n} ( \tau )) \  
+
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,
 
  \mathop{\rm exp}\nolimits \  \int\limits _ \tau  ^ t  \mathop{\rm Tr}\nolimits \  A (s) \  ds ,\quad\quad \tau ,\  t \in I,
 
$$
 
$$
Line 106: Line 103:
 
is applicable.
 
is applicable.
  
Let the functions (2) be the solutions of a linear homogeneous  $  n $-
+
Let the functions (2) be the solutions of a linear homogeneous  $  n $-th order equation
th order equation
 
  
 
$$  
 
$$  
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$$  
 
$$  
W (f _{1} (t) \dots f _{n} (t)) \quad \neq \quad 0,\quad\quad
+
W (f _{1} (t), \dots, f _{n} (t)) \quad \neq \quad 0,\quad\quad
 
t \in I.
 
t \in I.
 
$$
 
$$
Line 128: Line 124:
  
 
$$  
 
$$  
W (f _{1} (t) \dots f _{n} (t))\quad =
+
W (f _{1} (t), \dots, f _{n} (t))\quad =
 
$$
 
$$
  
Line 134: Line 130:
 
$$  
 
$$  
 
= \quad
 
= \quad
W (f _{1} ( \tau ) \dots f _{n} ( \tau )) \  \mathop{\rm exp}\nolimits
+
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,
 
\left [ - \int\limits _ \tau  ^ t p _{1} (s) \  ds \right ] ,\quad\quad \tau ,\  t \in I,
 
$$
 
$$
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A sub-Wronskian of order  $  i $
 
A sub-Wronskian of order  $  i $
for  $  \Phi = \{ f _{1} \dots f _{n} \} $
+
for  $  \Phi = \{ f _{1}, \dots, f _{n} \} $
 
is obtained by taking the Wronskian of a subset of size  $  i $
 
is obtained by taking the Wronskian of a subset of size  $  i $
 
of  $  \Phi $.  
 
of  $  \Phi $.  
 
Two theorems giving sufficient conditions for linear dependence in terms of Wronskians are as follows. 1) Let  $  n> 1 $,  
 
Two theorems giving sufficient conditions for linear dependence in terms of Wronskians are as follows. 1) Let  $  n> 1 $,  
$  f _{1} \dots f _{n} $
+
$  f _{1}, \dots, f _{n} $
 
analytic and  $  W( \Phi ) \equiv 0 $,  
 
analytic and  $  W( \Phi ) \equiv 0 $,  
then the  $  f _{1} \dots f _{n} $
+
then the  $  f _{1}, \dots, f _{n} $
 
are linearly dependent, [[#References|[a4]]], [[#References|[a5]]]. 2) Let  $  n > 1 $,  
 
are linearly dependent, [[#References|[a4]]], [[#References|[a5]]]. 2) Let  $  n > 1 $,  
 
$  W( \Phi ) \equiv 0 $,  
 
$  W( \Phi ) \equiv 0 $,  
but at no point of the interval of definition of  $  f _{1} \dots f _{n} $
+
but at no point of the interval of definition of  $  f _{1}, \dots, f _{n} $
 
do all sub-Wronskians of order  $  n - 1 $
 
do all sub-Wronskians of order  $  n - 1 $
 
vanish simultaneously, then  $  \Phi $
 
vanish simultaneously, then  $  \Phi $

Revision as of 12:03, 21 March 2022


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)


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
How to Cite This Entry:
Wronskian. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wronskian&oldid=44338
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article