In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Wronskian (or Wrońskian) is a
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
introduced by and named by . It is used in the study of
differential equations
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
, where it can sometimes show
linear independence
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
in a set of solutions.
Definition
The Wronskian of two differentiable functions and is .
More generally, for
real
Real may refer to:
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Music Albums
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* ''Real'' (Bright album) (2010) ...
- or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
-valued functions , which are times
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
on an
interval , the Wronskian as a function on is defined by
That is, it is the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of the
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
constructed by placing the functions in the first row, the first derivative of each function in the second row, and so on through the th derivative, thus forming a
square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
.
When the functions are solutions of a
linear differential equation
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
:a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b ...
, the Wronskian can be found explicitly using
Abel's identity
In mathematics, Abel's identity (also called Abel's formula or Abel's differential equation identity) is an equation that expresses the Wronskian of two solutions of a homogeneous second-order linear ordinary differential equation in terms of a c ...
, even if the functions are not known explicitly.
The Wronskian and linear independence
If the functions are linearly dependent, then so are the columns of the Wronskian (since differentiation is a linear operation), and the Wronskian vanishes. Thus, the Wronskian can be used to show that a set of differentiable functions is
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
on an interval by showing that it does not vanish identically. It may, however, vanish at isolated points.
A common misconception is that everywhere implies linear dependence, but pointed out that the functions and have continuous derivatives and their Wronskian vanishes everywhere, yet they are not linearly dependent in any neighborhood of . There are several extra conditions that ensure that the vanishing of the Wronskian in an interval implies linear dependence.
Maxime Bôcher
Maxime Bôcher (August 28, 1867 – September 12, 1918) was an American mathematician who published about 100 papers on differential equations, series, and algebra. He also wrote elementary texts such as ''Trigonometry'' and ''Analytic Geometry''. ...
observed that if the functions are
analytic, then the vanishing of the Wronskian in an interval implies that they are linearly dependent.
gave several other conditions for the vanishing of the Wronskian to imply linear dependence; for example, if the Wronskian of functions is identically zero and the Wronskians of of them do not all vanish at any point then the functions are linearly dependent. gave a more general condition that together with the vanishing of the Wronskian implies linear dependence.
Over fields of positive characteristic the Wronskian may vanish even for linearly independent polynomials; for example, the Wronskian of and 1 is identically 0.
Application to linear differential equations
In general, for an
th order linear differential equation, if
solutions are known, the last one can be determined by using the Wronskian.
Consider the second order differential equation in
Lagrange's notation
In differential calculus, there is no single uniform notation for differentiation. Instead, various notations for the derivative of a function or variable have been proposed by various mathematicians. The usefulness of each notation varies with ...
where
,
are known. Let us call
the two solutions of the equation and form their Wronskian
Then differentiating
and using the fact that
obey the above differential equation shows that
Therefore, the Wronskian obeys a simple first order differential equation and can be exactly solved:
where
and
is a constant.
Now suppose that we know one of the solutions, say
. Then, by the definition of the Wronskian,
obeys a first order differential equation:
and can be solved exactly (at least in theory).
The method is easily generalized to higher order equations.
Generalized Wronskians
For functions of several variables, a generalized Wronskian is a determinant of an by matrix with entries (with ), where each is some constant coefficient linear partial differential operator of order . 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
In mathematics, Roth's theorem is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that algebraic numbers cannot have many rational number approximations that are 'very good'. Over half ...
. For more general conditions under which the converse is valid see .
See also
*
Variation of parameters
In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations.
For first-order inhomogeneous linear differential equations it is usually possible t ...
*
Moore matrix In linear algebra, a Moore matrix, introduced by , is a matrix defined over a finite field. When it is a square matrix its determinant is called a Moore determinant (this is unrelated to the Moore determinant of a quaternionic Hermitian matrix). Th ...
, analogous to the Wronskian with differentiation replaced by the
Frobenius endomorphism
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphis ...
over a finite field.
*
Alternant matrix
In linear algebra, an alternant matrix is a matrix formed by applying a finite list of functions pointwise to a fixed column of inputs. An alternant determinant is the determinant of a square alternant matrix.
Generally, if f_1, f_2, \dots, f_n ...
*
Vandermonde matrix
In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row: an matrix
:V=\begin
1 & x_1 & x_1^2 & \dots & x_1^\\
1 & x_2 & x_2^2 & \dots & x_2^\\
1 & x_ ...
Notes
Citations
References
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{{Matrix classes
Ordinary differential equations
Determinants
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