View source for 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070660/o0706601.png" />-matrix, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070660/o0706602.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070660/o0706603.png" /> is the number of equations and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070660/o0706604.png" /> 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070660/o0706605.png" /></td> </tr></table>

is solvable if and only if the ranks of the basic matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070660/o0706606.png" /> and the expanded matrix obtained by adding the column of free terms to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070660/o0706607.png" /> coincide.

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070660/o0706608.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070660/o0706609.png" /> is a polynomial in one variable (an ordinary differential equation) or several variables (a partial differential equation), while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070660/o07066010.png" /> is the symbol of differentiation, the compatibility condition is a homogeneous system of equations with constant coefficients:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070660/o07066011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070660/o07066012.png" /> can be algebraically determined from the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070660/o07066013.png" />.

If (1) is a system of partial differential equations with variable coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070660/o07066014.png" />, it is much more difficult to determine compatibility conditions, which take the form (2) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070660/o07066015.png" />.

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070660/o07066016.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070660/o07066017.png" /></td> </tr></table>

An analytic function of several complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070660/o07066018.png" /> can also be considered as a solution to the overdetermined system of equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070660/o07066019.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070660/o07066020.png" />.

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



====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]].