Hypermatrix

Hypermatrix is a generalization of a matrix to higher dimensions, defined as a d-dimensional array A with entries a_{i1,...,id} where i_k ranges from 1 to n_k. When d equals 2, a hypermatrix reduces to a standard matrix.

In many mathematical contexts, the term hypermatrix is used interchangeably with tensor, especially when focusing on its coordinate representation. A tensor is the multilinear object independent of basis, while the hypermatrix of its components depends on the chosen basis and transforms according to the tensor transformation laws under a basis change.

Operations on hypermatrices include elementwise arithmetic, reshaping into matrices by matricization, and mode-n products with a matrix. Contractions along indices produce lower-order arrays, and outer/tensor products combine multiple arrays. Rank notions extend beyond the matrix case; common ones are CP-rank (the minimum number of rank-one outer products) and the multilinear (Tucker) rank given by the ranks of mode-n unfoldings. Computing tensor rank is generally NP-hard.

A hypermatrix is specified by its dimensions n1, n2, ..., nd and its entries a_{i1,...,id}. For example, a 3-by-4-by-5 hypermatrix has three axes and 60 entries. Such objects naturally arise when representing multi-way data or multi-modal measurements.

Applications span data analysis, machine learning, signal processing, chemometrics, psychometrics, and computer vision. Tensor decompositions, notably CANDECOMP/PARAFAC (CP) and Tucker decompositions, are used to extract latent structures and compress information in high-dimensional datasets.

See also: tensor, matrix, tensor decomposition, multilinear algebra, matricization.