propertiesnonnegativity

Nonnegativity is the property of being greater than or equal to zero. It is used across real numbers, vectors, matrices, functions, measures, and other mathematical objects. An object with nonnegativity is described as nonnegative.

For real numbers, nonnegative numbers satisfy closure under addition and multiplication: if a ≥ 0 and b ≥ 0 then a + b ≥ 0 and ab ≥ 0. The sum or product of nonnegative quantities remains nonnegative, and zero acts as a neutral element for both addition and multiplication. If a ≥ 0 and c is any real number, then c a is nonnegative only when c ≥ 0; otherwise ca ≤ 0.

For functions, a function f is nonnegative on a domain if f(x) ≥ 0 for all x in the domain. The sum or nonnegative scalar multiple of nonnegative functions is nonnegative; integrals or sums of nonnegative functions are nonnegative as well.

In linear algebra and optimization, a nonnegative matrix has all entries ≥ 0, and a nonnegative vector has all components ≥ 0. Nonnegativity constraints are common in optimization, reflecting quantities such as costs, counts, or probabilities. The product of nonnegative matrices is nonnegative, and norms or measures are nonnegative by definition.

Common examples include the square of any real number, x^2 ≥ 0, and the absolute value |x| ≥ 0; probability values lie in the interval [0, 1]. Nonnegativity is distinct from positivity, as nonnegative quantities may include zero. See also nonnegative matrices, nonnegative optimization, and probability theory.