nonnegativity

Nonnegativity is the property of being greater than or equal to zero. In mathematics, a quantity is nonnegative if its value lies in the interval [0, ∞). This notion is used across disciplines to describe quantities that cannot be negative, such as lengths, probabilities, and many physical or statistical measures.

Nonnegative numbers include zero and all positive numbers. The set of nonnegative real numbers is denoted [0, ∞). For many structured contexts, one distinguishes nonnegative elements in various sets, such as nonnegative integers, nonnegative matrices, or nonnegative random variables.

In functions, a function f is nonnegative on a domain D if f(x) ≥ 0 for all x in D. Nonnegative functions arise in optimization, analysis, and probability. If f ≥ 0 almost everywhere, its integral is nonnegative.

In linear algebra, a matrix with all entries nonnegative is called a nonnegative matrix and plays a role in Markov processes and Perron–Frobenius theory. A symmetric matrix A is nonnegative definite (often called positive semidefinite) when x^T A x ≥ 0 for all x. These notions constrain quadratic forms and have algorithmic and geometric implications.

Nonnegativity constraints are common in optimization problems, requiring decision variables to be nonnegative. They appear in linear programming, quadratic programming, and nonnegative least squares. In probability and statistics, nonnegativity is a natural property of quantities such as counts and probabilities, which are necessarily nonnegative.