Wronskian#The Wronskian and linear independence

The Wrońskian of two differentiable functions {{math|f }} and {{math|g}} is $$

W(f,g)=f g' - g f'

.

More generally, for {{math|n}} real- or complex-valued functions {{math|f₁, …, f_n}}, which are {{math|n – 1}} times differentiable on an interval {{math|I}}, the Wronskian $$

W(f_1,\ldots,f_n)

is a function on $$

x\in I

defined by

W(f_1, \ldots, f_n) (x)=

\det

\begin{bmatrix}

f_1(x) & f_2(x) & \cdots & f_n(x) \\

f_1'(x) & f_2'(x) & \cdots & f_n' (x)\\

\vdots & \vdots & \ddots & \vdots \\

f_1^{(n-1)}(x)& f_2^{(n-1)}(x) & \cdots & f_n^{(n-1)}(x)

\end{bmatrix}.

This is the determinant of the matrix constructed by placing the functions in the first row, the first derivatives of the functions in the second row, and so on through the $$

(n-1)^{\text{th}}

derivative, thus forming a square matrix.

When the functions {{math|f_i}} are solutions of a linear differential equation, the Wrońskian can be found explicitly using Abel's identity, even if the functions {{math|f_i}} are not known explicitly. (See below.)

If the functions {{math|f_i}} are linearly dependent, then so are the columns of the Wrońskian (since differentiation is a linear operation), and the Wrońskian vanishes. Thus, one may show that a set of differentiable functions is linearly independent on an interval by showing that their Wrońskian does not vanish identically. It may, however, vanish at isolated points.{{Citation

| last1 = Bender

| first1 = Carl M.

| author-link = Carl M. Bender

| last2 = Orszag

| first2 = Steven A.

| author2-link = Steven Orszag

| title = Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory

| place = New York

| publisher = Springer

| orig-year = 1978

| year = 1999

| page = 9

| isbn = 978-0-387-98931-0

}}

A common misconception is that {{math|1=W = 0}} everywhere implies linear dependence. {{harvtxt|Peano|1889}} pointed out that the functions {{math|x²}} and {{math|{{abs|x}}{{middot}} x}} have continuous derivatives and their Wrońskian vanishes everywhere, yet they are not linearly dependent in any neighborhood of {{math|0}}.{{efn|Peano published his example twice, because the first time he published it, an editor, Paul Mansion, who had written a textbook incorrectly claiming that the vanishing of the Wrońskian implies linear dependence, added a footnote to Peano's paper claiming that this result is correct as long as neither function is identically zero. Peano's second paper pointed out that this footnote was nonsense.}} There are several extra conditions which combine with vanishing of the Wronskian in an interval to imply linear dependence.

• Maxime Bôcher observed that if the functions are analytic, then the vanishing of the Wrońskian in an interval implies that they are linearly dependent.
• {{harvtxt|Bôcher|1901}} gave several other conditions for the vanishing of the Wrońskian to imply linear dependence; for example, if the Wrońskian of {{math|n}} functions is identically zero and the {{math|n}} Wrońskians of {{math|n – 1}} of them do not all vanish at any point then the functions are linearly dependent.
• {{harvtxt|Wolsson|1989a}} gave a more general condition that together with the vanishing of the Wronskian implies linear dependence.

Over fields of positive characteristic {{math|p}} the Wronskian may vanish even for linearly independent polynomials; for example, the Wronskian of {{math|x{{i sup|p}}}} and 1 is identically 0.

In general, for an $n$th order linear differential equation, if $(n-1)$ solutions are known, the last one can be determined by using the Wronskian.

Consider the second order differential equation in Lagrange's notation:

$$y\text{'}\text{'}\; =\; a(x)y\text{'}\; +\; b(x)y$$

where $a(x)$, $b(x)$ are known, and y is the unknown function to be found. Let us call $y\_1,\; y\_2$ the two solutions of the equation and form their Wronskian

$$W(x)\; =\; y\_1\; y\text{'}\_2\; -\; y\_2\; y\text{'}\_1$$

Then differentiating $W(x)$ and using the fact that $y\_i$ obey the above differential equation shows that

$$W\text{'}(x)\; =\; a(x)\; W(x)$$

Therefore, the Wronskian obeys a simple first order differential equation and can be exactly solved:

$$W(x)\; =\; C~e^\{A(x)\}$$

where $A\text{'}(x)=a(x)$ and $C$ is a constant.

Now suppose that we know one of the solutions, say $y\_2$. Then, by the definition of the Wrońskian, $y\_1$ obeys a first order differential equation:

$$y\text{'}\_1\; -\backslash frac\{y\text{'}\_2\}\{y\_2\}\; y\_1\; =\; -W(x)/y\_2$$

and can be solved exactly (at least in theory).

The method is easily generalized to higher order equations.

The relationship between the Wronskian and linear independence can also be strengthened in the context of a differential equation. If we have $n$ linearly independent functions that are all solutions of the same monic $n$th-order homogeneous-linear ordinary differential equation $y^\{(n)\}+Ly=0$ (where $L$ is a linear differential operator with respect to $x$ of order less than $n$) on some interval $I$, then their Wronskian is zero nowhere on $I$. Thus, counterexamples like $x^2$ and $x$

For {{math|n}} functions of several variables, a generalized Wronskian is a determinant of an {{math|n}} by {{math|n}} matrix with entries {{math|D_i(f_j)}} (with {{math|0 ≤ i < n}}), where each {{math|D_i}} is some constant coefficient linear partial differential operator of order {{math|i}}. If the functions are linearly dependent then all generalized Wronskians vanish. As in the single variable case the converse is not true in general: if all generalized Wronskians vanish, this does not imply that the functions are linearly dependent. However, the converse is true in many special cases. For example, if the functions are polynomials and all generalized Wronskians vanish, then the functions are linearly dependent. Roth used this result about generalized Wronskians in his proof of Roth's theorem. For more general conditions under which the converse is valid see {{harvtxt|Wolsson|1989b}}.

The Wrońskian was introduced by {{harvs|txt|authorlink=Józef Maria Hoene-Wroński|first=Józef|last=Hoene-Wroński|year=1812}} and given its current name by {{harvs|txt|authorlink=Thomas Muir (mathematician)|first=Thomas|last=Muir|year=1882|loc=Chapter XVIII}}.

• Variation of parameters
• Moore matrix, analogous to the Wrońskian with differentiation replaced by the Frobenius endomorphism over a finite field.
• Alternant matrix
• Vandermonde matrix

{{notelist}}

{{Reflist|refs=

{{cite journal |last1=Engdahl |first1=Susannah |last2=Parker |first2=Adam |title=Peano on Wronskians: A Translation |journal=Convergence |date=April 2011 |doi=10.4169/loci003642 |doi-access=free |url=https://www.maa.org/press/periodicals/convergence/peano-on-wronskians-a-translation |access-date=2020-10-08 |publisher=Mathematical Association of America |doi-broken-date=2024-11-12 |at=Section [https://www.maa.org/press/periodicals/convergence/peano-on-wronskians-a-translation-on-the-wronskian-determinant "On the Wronskian Determinant"] |quote=The most famous theorem is attributed to Bocher, and states that if the Wronskian of $n$ analytic functions is zero, then the functions are linearly dependent ([B2], [BD]). [The citations 'B2' and 'BD' refer to {{harvs|txt|last=Bôcher|year=1900–1901}} and {{harvs|txt|last=Bostan|last2=Dumas|year=2010}}, respectively.]}}

{{cite journal |last1=Engdahl |first1=Susannah |last2=Parker |first2=Adam |title=Peano on Wronskians: A Translation |journal=Convergence |date=April 2011 |doi=10.4169/loci003642 |doi-access=free |url=https://www.maa.org/press/periodicals/convergence/peano-on-wronskians-a-translation |access-date=2020-10-08 |publisher=Mathematical Association of America|doi-broken-date=2024-11-12 }}

}}

• {{cite journal | last=Bôcher | first=Maxime | author-link=Maxime Bôcher | title=The Theory of Linear Dependence | journal=Annals of Mathematics | publisher=Princeton University | volume=2 | issue=1/4 | year=1900–1901 | issn=0003-486X | doi=10.2307/2007186 | doi-access=free | jstor=2007186 | jstor-access=free | pages=81–96| hdl=2027/hvd.hn57mn | hdl-access=free }}
• {{Citation | last1=Bôcher | first1=Maxime | title=Certain cases in which the vanishing of the Wronskian is a sufficient condition for linear dependence | jstor=1986214 | jstor-access=free | publisher=American Mathematical Society | location=Providence, R.I. | year=1901 | journal=Transactions of the American Mathematical Society | issn=0002-9947 | volume=2 | issue=2 | pages=139–149 | jfm=32.0313.02 | doi=10.2307/1986214 | doi-access=free | url=https://www.ams.org/journals/tran/1901-002-02/S0002-9947-1901-1500560-5/S0002-9947-1901-1500560-5.pdf }}
• {{cite journal | last1=Bostan | first1=Alin |first2=Philippe |last2=Dumas | title=Wronskians and Linear Independence | journal=American Mathematical Monthly | publisher=Taylor & Francis | volume=117 | issue=8 | year=2010 | issn=0002-9890 | doi=10.4169/000298910x515785 | jstor=10.4169/000298910x515785 | pages=722–727 | arxiv=1301.6598 | s2cid=9322383 }}
• {{Citation | last1=Hartman | first1=Philip | title=Ordinary Differential Equations | url=https://books.google.com/books?id=CENAPMUEpfoC | publisher=John Wiley & Sons | location=New York | isbn=978-0-89871-510-1 | mr=0171038 | zbl=0125.32102 | year=1964}}
• {{citation|first=Józef |last=Hoene-Wroński|title=Réfutation de la théorie des fonctions analytiques de Lagrange|place= Paris |year=1812}}
• {{Citation | last1=Muir | first1=Thomas | title=A Treatise on the Theorie of Determinants. | url=https://archive.org/details/atreatiseontheo00muirgoog | publisher= Macmillan | year=1882 | jfm=15.0118.05}}
• {{Citation | last1=Peano | first1=Giuseppe | author1-link=Giuseppe Peano | title=Sur le déterminant wronskien. | language=fr | jfm=21.0153.01 | year=1889 | journal=Mathesis | volume=IX | pages=75–76, 110–112|url=http://www.maa.org/press/periodicals/convergence/peano-on-wronskians-a-translation-appendix-2-translations}}
• {{eom|title=Wronskian|first=N. Kh. |last=Rozov}}
• {{Citation | last1=Wolsson | first1=Kenneth | title=A condition equivalent to linear dependence for functions with vanishing Wronskian | doi=10.1016/0024-3795(89)90393-5 | mr=989712 | zbl=0671.15005 | year=1989a | journal=Linear Algebra and Its Applications | issn=0024-3795 | volume=116 | pages=1–8| doi-access=free }}
• {{Citation | last1=Wolsson | first1=Kenneth | title=Linear dependence of a function set of {{math|m}} variables with vanishing generalized Wronskians | doi=10.1016/0024-3795(89)90548-X | mr=993032 | zbl=0724.15004 | year=1989b | journal=Linear Algebra and Its Applications | issn=0024-3795 | volume=117 | pages=73–80| doi-access=free }}

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