Underdimensionerat

Underdimensionerat is a term used in Swedish-language discourse to describe models, systems or problems that have fewer independent constraints than needed to determine a unique outcome. In English academic usage the corresponding concepts are underdetermined or underconstrained. The term is commonly applied in mathematics, engineering, statistics and data science, where the information available does not fully specify all unknowns without additional assumptions.

In linear algebra, an underdetermined system occurs when a matrix A has more unknowns than equations: Ax = b with m < n. If the rank r of A is less than n but b lies in the column space of A, the solution set is infinite, forming a linear manifold of dimension n − r. Without extra criteria, many different x satisfy Ax = b.

Practically, one uses regularization or prior information to select a unique solution. Techniques include the Moore–Penrose pseudoinverse for a least-norm solution, Tikhonov ridge regularization, or sparsity-promoting methods in compressed sensing. The choice of criterion reflects domain knowledge or desired properties such as smoothness or minimal energy.

Underdimensionerat problems are widespread in data acquisition and modeling. They occur when there are more variables than measurements, when data are incomplete, or when models omit relevant factors. Recognizing underdimensionering helps guide the use of regularization, extra data collection, or model reformulation to achieve identifiability.