Nonpositive

Nonpositive is a mathematical term describing quantities that are not positive. The most common usage refers to real numbers that are less than or equal to zero. The set of nonpositive real numbers is (-∞, 0], consisting of zero and all negative numbers. This set is the complement of the nonnegative numbers on the real line, which are [0, ∞).

In analysis, a function is called nonpositive if its values are never positive, meaning f(x) ≤ 0 for all x in its domain. Similarly, a sequence or a vector-valued function is nonpositive if each of its terms or components is ≤ 0.

In linear algebra and matrix analysis, the term appears in phrases such as nonpositive definite. However, the standard terminology for a symmetric matrix A that satisfies x^T A x ≤ 0 for all vectors x is negative semidefinite. Some authors use nonpositive to emphasize the absence of any positive quadratic form, but negative semidefinite is generally preferred for precision.

In optimization and related fields, nonpositivity conditions arise as constraints or properties of objective functions or residuals, indicating that certain quantities do not increase above zero. The term serves as a general negation of positivity across contexts, from numbers to functions and, with more specialized meaning, to matrices and quadratic forms.