---
tag: 9R899M1746
kind: definition
name: Overdetermined system
msc: [15A06, 65F20]
defines: QJ0P26J9BV
uses: [0JTRMBQ1SA]
sources:
  - url: https://en.wikipedia.org/wiki/Overdetermined_system
---

## Definition

Let $F$ be a [[field|04C5ZBG1WT]].
Let $A$ be an $m \times n$ matrix over $F$ and let $b \in F^m$.
The [[system of linear equations|0GT4WWGWH2]] $Ax = b$ in the unknown $x \in F^n$ is *overdetermined* if $m > n$, that is, if it has more equations than unknowns.

## Remarks

Example: the system $x = 1$, $y = 2$, $x + y = 3$ over $\R$ has $m = 3$ equations and $n = 2$ unknowns, so it is overdetermined; it is consistent, with the solution $(x, y) = (1, 2)$.
Non-example: the system $x + y = 1$, $x - y = 0$ has $m = n = 2$, so it is not overdetermined.
If $m > n$, then the [[column space|03AJKXH1AV]] of $A$ has dimension at most $n < m$, so it is a proper subspace of $F^m$.
Hence there are right-hand sides $b$ for which the system is [[inconsistent|0X6V9B1V10]].
Over $\R$, the consistent right-hand sides form a proper subspace of $\R^m$, so a randomly chosen $b$ almost never gives a consistent system.
Over $\R$, the [[least squares|0BYTQ4X1HC]] solutions are the solutions of the normal equations $A^{T} A x = A^{T} b$, which are always consistent.
A system with fewer equations than unknowns is called *underdetermined*.