matrixpossibly

Matrixpossibly is a concept in mathematics and computer science used to describe a matrix in which each entry represents a set of possible values rather than a single fixed value. This approach encodes uncertainty, variability, or incomplete information in a compact matrix form, enabling analysis that accounts for multiple potential outcomes within a unified structure.

Formally, a matrixpossibly M can be viewed as a function M: I × J → P(D), where I and J index the rows and columns, D is a value domain, and P(D) denotes the power set of D (the set of all possible values for each entry). In practice, entries may also be represented as intervals, probability distributions, or other compact encodings of uncertainty. Matrixpossibles are closely related to interval matrices, fuzzy matrices, and probabilistic matrices, but they emphasize the notion of a set of potential values per entry.

Operations on matrixpossibles extend standard matrix arithmetic to set-valued entries. For example, the elemental operation A ⊕ B may be defined entrywise as combining all pairs (a, b) with a ∈ A_ij and b ∈ B_ij to form the result set under the chosen rule (such as union, interval hull, or a probabilistic aggregation). Reduction to a single-valued matrix can be achieved through defuzzification, expectation, or choosing representative values, depending on the application.

Applications of matrixpossibles appear in robust optimization, scenario analysis, and uncertainty quantification, where representing multiple potential values per cell aids in evaluating sensitivity and worst-case versus best-case outcomes. See also interval matrices, fuzzy matrices, and probabilistic matrices for related formalisms.