Overdetermined

In mathematics, an overdetermined system is one that contains more equations than unknowns. This situation commonly arises as m equations in n unknowns with m > n. In linear algebra, this is written as Ax = b where A is an m by n matrix, x is an n-vector of unknowns, and b is an m-vector. If the matrix A has full column rank (rank n), the system is generally inconsistent for arbitrary b: there is typically no exact solution. When b lies in the column space of A, an exact solution exists and is unique, but such cases are special.

A simple illustration is two unknowns x and y with three equations, such as x + y = 2, x − y = 0, and 2x + 2y = 4. Depending on the equations, the system may be consistent with a unique solution (x = 1, y = 1) or inconsistent if one equation contradicts the others (for example, x + y = 3 instead of 2).

Because exact solutions are often unavailable, overdetermined systems are typically solved by least-squares methods, which seek the x that minimizes the residual ||Ax − b||. The least-squares solution is x̂ = (AᵀA)⁻¹Aᵀb when A has full column rank; more generally, the solution is given by the Moore–Penrose pseudoinverse A⁺.

In statistics and data analysis, an overdetermined model occurs when there are more observations than parameters, leading to regression problems. In differential equations and geometry, overdetermined systems may impose compatibility (integrability) conditions for a solution to exist. They are commonly contrasted with underdetermined systems (fewer equations than unknowns) and with determined systems (exactly as many independent equations as unknowns).