Matriceje

Matriceje is a concept in multilinear algebra describing a class of multi-dimensional arrays that generalize matrices. A matriceje of order k is an array A with dimensions n1×n2×...×nk whose entries lie in a field. The key operation, the matricial product, generalizes matrix multiplication by performing contractions over shared indices.

Through a prescribed pattern of contractions, two matriceje can be combined to yield a new object whose order is either reduced or reshaped, depending on the contraction pattern. For a special case when k=2, the matriceje reduces to ordinary matrices and their product coincides with standard matrix multiplication.

Matriceje form an algebra under the matricial product, with an identity object corresponding to a diagonal array of appropriate dimensions. Many basic linear algebra notions extend notionally: rank, invertibility, and invariants such as generalized determinants or hyperdeterminants can be defined in certain frameworks.

In practice, to compute with matriceje, one commonly unfolds (matricizes) the array along a mode to obtain a conventional matrix, apply standard algorithms, and, if needed, reassemble the results.

Applications are found in higher-order data analysis, quantum information, and machine learning, where multi-way interactions are important. Theoretical work on matriceje also explores questions of existence, stability under contraction, and relationships to other tensor frameworks such as tensors and tensor networks.

The term matriceje is etymologically related to matrix plus a Slavic plural suffix, and its use remains mostly in theoretical discussions and niche computational frameworks. See also: matrices, tensors, multilinear algebra, tensor contractions.