Difference between revisions of "Wronskian"
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''Wronski determinant'' | ''Wronski determinant'' | ||
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of the type | 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 [[#References|[1]]]. | The concept was first introduced by J. Wronski [[#References|[1]]]. | ||
− | If the vector-functions (1) are linearly dependent on a set | + | 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 | ||
+ | |||
+ | $$ | ||
+ | W ( \phi _{1} (t) \dots \phi _{n} (t) ) \quad \neq \quad 0,\quad\quad | ||
+ | t \in I. | ||
+ | $$ | ||
+ | |||
− | + | 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. | applies. | ||
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====Comments==== | ====Comments==== | ||
− | An example of | + | 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 | + | 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]]]. |
Revision as of 13:40, 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 |
Wronskian. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wronskian&oldid=44338