Talk:Wronskian

Does the Wronskian need to vanish at every point?[edit]

f1(x) = x, f2(x) = x^2, are linearly independent.

$W(f_{1},f_{2})={\begin{vmatrix}x&x^{2}\\1&2x\end{vmatrix}}=2x^{2}-x^{2}=x^{2}$

W(f1, f2) is 0 at x=0, but it doesn't matter, because we only need one point where W is non-zero. x=1, W=1 is non-zero, so the functions are linearly independent.

The converse is not necessarily true: W=0 everywhere does not imply that the functions are dependent. Example: f1 = x^3, f2 = abs(x^3). They are linearly independent but the Wronskian is uniformly zero.

Dzhim 02:10, Feb 4, 2005 (UTC)

You're absolutely right. I thought about it for some time before I reverted you, but now I don't know what I was thinking. Sorry about that. -- Jitse Niesen 11:56, 4 Feb 2005 (UTC)

The converse is true if the two functions are real, differentiable and have only simple zero in the interval; or if the two functions are analytic

Davide

Linear dependence at a point[edit]

I removed the sentence about linear dependence at a point. As it stoodthat didn't say anything (for n > 1 you will always have dependence at a given point).

Charles Matthews 06:49, 23 Aug 2003 (UTC)

Homogeneous comment[edit]

I removed the condition that "the functions are solutions of some homogenous linear differential equation" from the statement "If the Wronskian is non-zero at some point in an interval, then the functions are linearly independent in that interval." If the Wronskian is nonzero at some point x, then the vectors

[f_{k},f'_{k},\ldots ,f_{k}^{(n-1)}],\quad k=1,\ldots ,n

are linearly independent, so every linear combination of the vectors will be a nonzero vector, so every linear combination of f_1, ..., f_n will have a nonzero derivative at x, hence it is not zero in a neighbourhood of x. -- Jitse Niesen (talk) 13:01, 12 October 2005 (UTC)[reply]

On calculators[edit]

I moved the following text from the article.

"Wronskian on the HP 49 series

You can simply edit the equations into the hp49, just use the matrix editor to enter the equations into the 49 and then do the det command to work out the Wronskian of a set of equations, having said this there must be an easyier way and I am looking into making a small program to do this, if you are interested please send me a message! user:Mick8882003"

The first bit (how to evaluate the Wronskian) does not give much useful information; it is too specific and it is rather straightforward, in my opinion. The second bit clearly does not belong in an encyclopaedia. -- Jitse Niesen (talk) 13:45, 18 December 2005 (UTC)[reply]

Proof[edit]

The motto of the supplied proof on 'Wronskian and linear independence' is mechanical but true. I think Wikepedia should seriously consider allowing special PROOF subpages for mathematical articles.

Ben Spinozoan 08:08, 29 December 2005 (UTC)[reply]

This has been discussed before. My view: proofs here have to justify their place, like any other content. Charles Matthews 09:57, 29 December 2005 (UTC)[reply]

But anyway this proof is over the top, no? A linear dependence gives a vanishing determinant for more obvious reasons. The column rank is less than n. Charles Matthews 11:43, 29 December 2005 (UTC)[reply]

Touché.....I never said it was elegant! Be my guest and supplant at will. Ben Spinozoan 14:47, 29 December 2005 (UTC)[reply]

Actually... the proof is incorrect. Just because differentiation is a linear operation, it doesn't mean that linearly independent functions will map to linearly independent columns, since differentiation is a nontrivial kernel (the set of all constant functions); so, the proof needs some elaboration. —Preceding unsigned comment added by 18.243.2.126 (talk) 20:23, 6 October 2007 (UTC)[reply]

Would this be a proof? http://tutorial.math.lamar.edu/Classes/DE/FundamentalSetsofSolutions.aspx It makes sense to me. —Preceding unsigned comment added by 201.62.109.74 (talk) 20:58, 26 November 2007 (UTC)[reply]

Mathematical expression is garbled[edit]

Can someone please fix the syntax of the expression for the Wronskian in the third example in the article? It appears garbled right now. Chaimike (talk) 05:40, 22 February 2009 (UTC)[reply]

Further Examples[edit]

how about decaying exponentials? Do the functions change from being linearly independent after some arbitrarily large interval in time when the wronskian becomes 0? —Preceding unsigned comment added by 24.80.108.203 (talk) 22:47, 8 August 2010 (UTC)[reply]

Fundamental matrix[edit]

I am marking as dubious the uncited statement that the square matrix of which a Wrońskian is the determinant is "sometimes called a fundamental matrix". The sources I see (e.g. ^[1]^[2]^[3]^[4]) speak of a "fundamental matrix" specifically when solutions of differential equations are involved. Even if the Wrońskian is used only in that case, the definition given here is general enough that solutions of differential equations are not necessarily involved and the square matrix could be made from any sequence of sufficiently-smooth functions and their derivatives. Does someone else see sources that call any square matrix of functions and their derivatives that fits this pattern a "fundamental matrix"? —2d37 (talk) 04:36, 9 October 2020 (UTC)[reply]

I see that some authors (e.g. ^[5]) apparently refer to "the Wronskian matrix" and "the Wronskian determinant (or simply the Wronskian)" (emphasis sic). This seems pleasantly parallel to "Jacobian matrix" and "Jacobian determinant" and "Hessian matrix" and "Hessian determinant", but unfortunately referring in this way to Wrońskian matrices and determinants doesn't seem common. —2d37 (talk) 07:51, 9 October 2020 (UTC)[reply]

The claim in question was deleted (not by me) in Special:Diff/988405688. —2d37 (talk) 07:05, 9 December 2020 (UTC)[reply]

References

^ Komlenko, Yu.V. (2001) [1994], "Fundamental matrix", Encyclopedia of Mathematics, EMS Press
^ Walter, Wolfgang (2013). Ordinary Differential Equations. Graduate Texts in Mathematics. Translated by Thompson, Russell. New York: Springer Science & Business Media. p. 165. ISBN 978-1-4612-0601-9. Retrieved 2020-10-09.
^ Chicone, Carmen (2006). Ordinary Differential Equations with Applications. Texts in Applied Mathematics. New York: Springer Science & Business Media. p. 150. ISBN 978-0-387-30769-5. Retrieved 2020-10-09.
^ Vrabie, Ioan (2004). Differential Equations: An Introduction to Basic Concepts, Results and Applications. EBSCO ebook academic collection. World Scientific. p. 148. ISBN 978-981-238-838-4. Retrieved 2020-10-09.
^ Shilov, Georgi (2013). Elementary Functional Analysis. Dover Books on Mathematics. Dover Publications. p. 179. ISBN 978-0-486-31868-4. Retrieved 2020-10-09.

[EOM_Fundamental_matrix-1] Komlenko, Yu.V. (2001) [1994], "Fundamental matrix", Encyclopedia of Mathematics, EMS Press

[Thompson_Walter_2013_p._165-2] Walter, Wolfgang (2013). Ordinary Differential Equations. Graduate Texts in Mathematics. Translated by Thompson, Russell. New York: Springer Science & Business Media. p. 165. ISBN 978-1-4612-0601-9. Retrieved 2020-10-09.

[Chicone_2006_p._150-3] Chicone, Carmen (2006). Ordinary Differential Equations with Applications. Texts in Applied Mathematics. New York: Springer Science & Business Media. p. 150. ISBN 978-0-387-30769-5. Retrieved 2020-10-09.

[Vrabie_2004_p._148-4] Vrabie, Ioan (2004). Differential Equations: An Introduction to Basic Concepts, Results and Applications. EBSCO ebook academic collection. World Scientific. p. 148. ISBN 978-981-238-838-4. Retrieved 2020-10-09.

[Shilov_2013_p._179-5] Shilov, Georgi (2013). Elementary Functional Analysis. Dover Books on Mathematics. Dover Publications. p. 179. ISBN 978-0-486-31868-4. Retrieved 2020-10-09.

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