Overdetermined system

A system in which the number of equations is greater than the number of unknowns. In the linear case, such a system is given by a rectangular -matrix, , where is the number of equations and the number of unknowns. The primary question for an overdetermined system is its solvability, expressed by compatibility conditions.

For example, an overdetermined system of linear algebraic equations

is solvable if and only if the ranks of the basic matrix and the expanded matrix obtained by adding the column of free terms to coincide.

For an overdetermined system of linear differential equations with constant coefficients,

(1)

where is a polynomial in one variable (an ordinary differential equation) or several variables (a partial differential equation), while is the symbol of differentiation, the compatibility condition is a homogeneous system of equations with constant coefficients:

(2)

where the matrix can be algebraically determined from the matrix .

If (1) is a system of partial differential equations with variable coefficients , it is much more difficult to determine compatibility conditions, which take the form (2) with .

The simplest example of an overdetermined system is the system of differential equations

The compatibility conditions for this system, necessary and sufficient for it to be solvable, take the form

An analytic function of several complex variables can also be considered as a solution to the overdetermined system of equations

where .

References

[1]	A.I. Mal'tsev, "Foundations of linear algebra" , Freeman (1963) (Translated from Russian)
[2]	V.V. Palamodov, "Systems of linear differential equations" Progress in Math. , 10 (1971) pp. 1–36 Itogi Nauk. Mat. Anal. (1969) pp. 5–37

Comments

The best known overdetermined system of partial differential equations is the Cauchy-Riemann equations for analytic functions in $\mathbb C^n$ with $n>1$. See also Neumann problem.