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''Wronski determinant''
 
''Wronski determinant''
  
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of the type
 
of the type
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w0981804.png" /></td> </tr></table>
+
$$
 +
W ( \phi _{1} (t) \dots \phi _{n} (t)) \quad = \quad
 +
\left |
  
The Wronskian of a system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w0981805.png" /> scalar functions
+
\begin{array}{lll}
 +
\phi _{1} ^{1} (t)  &\dots  &\phi _{n} ^{1} (t)  \\
 +
\dots  &\dots  &\dots  \\
 +
\phi _{1} ^{n} (t)  &\dots  &\phi _{n} ^{n} (t)  \\
 +
\end{array}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w0981806.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\right | .
 +
$$
  
which have derivatives up to order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w0981807.png" /> (inclusive) is the determinant
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w0981808.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
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)) \quad = \quad
 +
\left |
 +
 
 +
\begin{array}{lll}
 +
f _{1} (t)  &\dots  &f _{n} (t)  \\
 +
f _{1} ^ {\  \prime} (t)  &\dots  &f _{n} ^ {\  \prime} (t)  \\
 +
\dots  &\dots  &\dots  \\
 +
f _{1} ^ {\  (n-1)} (t)  &\dots  &f _{n} ^ {\  (n-1)} (t) \\
 +
\end{array}
 +
 
 +
\right | .
 +
$$
 +
 
  
 
The concept was first introduced by J. Wronski [[#References|[1]]].
 
The concept was first introduced by J. Wronski [[#References|[1]]].
  
If the vector-functions (1) are linearly dependent on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w0981809.png" />, then
+
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|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
 +
 
 +
$$
 +
W ( \phi _{1} (t) \dots \phi _{n} (t) ) \quad \neq \quad 0,\quad\quad
 +
t \in I.
 +
$$
 +
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818010.png" /></td> </tr></table>
+
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
  
If the scalar functions (2) are linearly dependent on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818011.png" />, then
+
$$
 +
W ( \phi _{1} (t) \dots \phi _{n} (t) )\quad =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818012.png" /></td> </tr></table>
 
  
The converse theorems are usually not true: Identical vanishing of a Wronskian on some set is not a sufficient condition for [[Linear dependence|linear dependence]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818013.png" /> functions on this set.
+
$$
 +
= \quad
 +
W ( \phi _{1} ( \tau ) \dots \phi _{n} ( \tau )) \
 +
\mathop{\rm exp}\nolimits \  \int\limits _ \tau  ^ t  \mathop{\rm Tr}\nolimits \  A (s) \  ds ,\quad\quad \tau ,\  t \in I,
 +
$$
  
Let the vector-functions (1) be the solutions of a linear homogeneous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818014.png" />-th order system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818016.png" />, with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818017.png" />-dimensional matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818018.png" /> that is continuous on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818019.png" />. If these solutions constitute a fundamental system, then
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818020.png" /></td> </tr></table>
+
where  $  \mathop{\rm Tr}\nolimits \  A(t) $
 +
is the trace of the matrix  $  A(t) $,
 +
is applicable.
  
If the Wronskian of these solutions is equal to zero in at least one point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818021.png" />, it is identically equal to zero on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818022.png" />, and the functions (1) are linearly dependent. The Liouville formula
+
Let the functions (2) be the solutions of a linear homogeneous  $  n $-
 +
th order equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818023.png" /></td> </tr></table>
+
$$
 +
y ^{(n)} + p _{1} (t) y ^{(n-1)} + \dots + p _{n-1} (t) y ^ \prime  + p _{n} (t) y \quad = \quad 0
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818024.png" /></td> </tr></table>
 
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818025.png" /> is the trace of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818026.png" />, is applicable.
+
with continuous coefficients on the interval  $  I $.  
 +
If these solutions constitute a fundamental system, then
  
Let the functions (2) be the solutions of a linear homogeneous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818027.png" />-th order equation
+
$$
 +
W (f _{1} (t) \dots f _{n} (t)) \quad \neq \quad 0,\quad\quad
 +
t \in I.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818028.png" /></td> </tr></table>
 
  
with continuous coefficients on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818029.png" />. If these solutions constitute a fundamental system, then
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818030.png" /></td> </tr></table>
+
$$
 +
W (f _{1} (t) \dots f _{n} (t))\quad =
 +
$$
  
If the Wronskian of these solutions is zero in at least one point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818031.png" />, it is identically equal to zero on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818032.png" />, and the functions (2) are linearly dependent. The Liouville formula
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818033.png" /></td> </tr></table>
+
$$
 +
= \quad
 +
W (f _{1} ( \tau ) \dots f _{n} ( \tau )) \  \mathop{\rm exp}\nolimits
 +
\left [ - \int\limits _ \tau  ^ t p _{1} (s) \  ds \right ] ,\quad\quad \tau ,\  t \in I,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818034.png" /></td> </tr></table>
 
  
 
applies.
 
applies.
Line 63: Line 147:
  
 
====Comments====
 
====Comments====
An example of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818035.png" /> functions (2) that are not linearly dependent but with vanishing Wronskian was given by G. Peano, [[#References|[a3]]].
+
An example of $  n $
 +
functions (2) that are not linearly dependent but with vanishing Wronskian was given by G. Peano, [[#References|[a3]]].
  
A sub-Wronskian of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818037.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818038.png" /> is obtained by taking the Wronskian of a subset of size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818039.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818040.png" />. Two theorems giving sufficient conditions for linear dependence in terms of Wronskians are as follows. 1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818042.png" /> analytic and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818043.png" />, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818044.png" /> are linearly dependent, [[#References|[a4]]], [[#References|[a5]]]. 2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818046.png" />, but at no point of the interval of definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818047.png" /> do all sub-Wronskians of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818048.png" /> vanish simultaneously, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098180/w09818049.png" /> is linearly dependent, [[#References|[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, [[#References|[a4]]], [[#References|[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, [[#References|[a3]]].
  
 
For more information and results concerning functions of several variables, cf. [[#References|[a6]]], [[#References|[a7]]].
 
For more information and results concerning functions of several variables, cf. [[#References|[a6]]], [[#References|[a7]]].

Latest revision as of 14:40, 24 January 2020


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

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


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

$$ \tag{3} W (f _{1} (t) \dots f _{n} (t)) \quad = \quad \left | \begin{array}{lll} f _{1} (t) &\dots &f _{n} (t) \\ f _{1} ^ {\ \prime} (t) &\dots &f _{n} ^ {\ \prime} (t) \\ \dots &\dots &\dots \\ f _{1} ^ {\ (n-1)} (t) &\dots &f _{n} ^ {\ (n-1)} (t) \\ \end{array} \right | . $$


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://www.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