MxMatrix

MxMatrix is a theoretical construct used in discussions of hierarchical linear operators. It generalizes the concept of a matrix by organizing entries as matrices themselves, recursively, to form a multi-level block structure.

Notation and indexing: An n-by-m MxMatrix comprises blocks B_ij, where each block B_ij is itself a p_i_j by q_i_j matrix, and blocks may vary in size. This allows representing linear operators on spaces that are themselves decomposed into subspaces, such as block-structured discretizations in finite element methods or tensor product spaces.

Operations: MxMatrix supports addition and multiplication defined blockwise, using standard matrix arithmetic at the lowest level and respecting block dimension compatibility. Specialized variants include sparse MxMatrices, where many blocks are empty, and diagonal MxMatrices, where nonzero blocks lie on a diagonal.

Properties and theory: MxMatrices preserve linearity; under certain conditions, the product of two MxMatrices corresponds to composition of the underlying block maps. They can be represented compactly using nested data structures and are amenable to parallel computation.

Applications: used in theoretical investigations of hierarchical linear operators, simulations using multi-resolution discretizations, quantum information models with subsystem structure, and some machine-learning architectures that employ block-structured weight matrices.

History and variants: The concept is discussed in hypothetical or exploratory mathematical literature; practical implementations rely on software abstractions that treat MxMatrices as data containers for block matrices. Related concepts include block matrices, sparse matrices, hierarchical matrices, and tensor networks.