Difference between revisions of "Wronskian"
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\left | | \left | | ||
− | \begin{array}{ | + | \begin{array}{ccc} |
− | \phi _{1} ^{1} (t) &\ | + | \phi _{1} ^{1} (t) &\cdots &\phi _{n} ^{1} (t) \\ |
− | \ | + | \vdots &\ddots &\vdots \\ |
− | \phi _{1} ^{n} (t) &\ | + | \phi _{1} ^{n} (t) &\cdots &\phi _{n} ^{n} (t) \\ |
\end{array} | \end{array} | ||
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− | 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)) | + | W (f _{1} (t), \dots, f _{n} (t)) = |
\left | | \left | | ||
− | \begin{array}{ | + | \begin{array}{ccc} |
− | f _{1} (t) &\ | + | f _{1} (t) &\cdots &f _{n} (t) \\ |
− | f _{1} ^ {\ \prime} (t) &\ | + | f _{1} ^ {\ \prime} (t) &\cdots &f _{n} ^ {\ \prime} (t) \\ |
− | \ | + | \vdots &\ddots &\vdots \\ |
− | f _{1} ^ {\ (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 = |
$$ | $$ | ||
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$$ | $$ | ||
= \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, | ||
$$ | $$ | ||
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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. | ||
$$ | $$ | ||
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$$ | $$ | ||
− | W (f _{1} (t) \dots f _{n} (t))\quad = | + | W (f _{1} (t), \dots, f _{n} (t))\quad = |
$$ | $$ | ||
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$$ | $$ | ||
= \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 11:58, 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
$$ W ( \phi _{1} (t) \dots \phi _{n} (t)) \quad = \quad \left | \begin{array}{ccc} \phi _{1} ^{1} (t) &\cdots &\phi _{n} ^{1} (t) \\ \vdots &\ddots &\vdots \\ \phi _{1} ^{n} (t) &\cdots &\phi _{n} ^{n} (t) \\ \end{array} \right | . $$
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 ![]() |
Wronskian. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wronskian&oldid=44338