matriksis

Matriksis is a term used in theoretical discussions to denote a generalized, matrix-like object that extends the standard matrix concept to higher dimensions and more flexible multiplication rules. In this broad sense, a matriksis acts as an algebraic container for multi-dimensional data, with operations designed to resemble familiar matrix arithmetic while allowing interactions across multiple indices.

A matriksis of order k over a field F is a k-dimensional array A = (a i1 ... ik) with entries in F, equipped with addition defined elementwise. The defining feature is a contracted product that combines two matriksises along prescribed modes, guided by a contraction tensor or by a bilinear map. When k = 2 and the contraction reduces to ordinary summation over a shared index, the matriksis reduces to the ordinary matrix product. More generally, the construction yields a multimodal multiplication that generalizes matrix multiplication to higher dimensions.

Properties vary by formulation. Some approaches assume associativity of the product and the existence of an identity element, yielding a matriksis algebra. Others relax these constraints, focusing on multilinear invariants or spectral-like properties. Determinants are not universally defined; alternative invariants or eigenvalue notions for matriksises may be used instead.

Applications of matriksis concepts appear in theoretical multilinear algebra, higher-order data analysis, and some science fiction or speculative mathematics where multi-way transformations are needed. They serve as a conceptual tool to reason about how data or operators behave when extended beyond two dimensions.

Because matriksis is not a single standardized notion, definitions and axioms differ across sources. Readers encountering the term should consult the specific formalism in use for precise rules and properties.