Reducedform

Reduced form is a term used in a variety of mathematical contexts to denote a standardized or simplest representation of a structure or quantity. The precise meaning depends on the field, but the core idea is to replace a given object with a form that is easier to analyze, compare, or compute with.

In linear algebra, reduced row-echelon form (RREF) is a canonical form of a matrix obtained by Gaussian elimination. A matrix is in RREF if every nonzero row has a leading 1, the leading 1 in each row is to the right of the one in the previous row, each leading 1 is the only nonzero entry in its column, and all zero rows lie at the bottom. RREF reveals the rank and the dependency relations among columns and simplifies solving linear systems.

In arithmetic, a fraction is in reduced form (lowest terms) when the numerator and denominator are coprime, i.e., their greatest common divisor is 1, and the denominator is positive. For example, the fraction 6/15 reduces to 2/5.

In econometrics and statistics, reduced-form models express endogenous variables directly as functions of exogenous variables, obtained by solving the structural equations. They are convenient for prediction and for studying the total effect of exogenous shocks, but they obscure causal structure present in the structural form.

In number theory and the theory of quadratic forms, reduced forms provide canonical representatives of equivalence classes under a symmetry group, such as SL2(Z) in binary quadratic forms. The reduced form rules give a standard way to classify and compare forms.