Matrylinearny

Matrylinearny is a term used in some Polish mathematical contexts to describe a structure that combines matrix-like organization with linearity in a hierarchical, nested form. The name evokes the idea of matryoshka dolls, signaling multiple levels of linear transformations embedded within a single object.

Formally, one can consider a sequence of vector spaces V0, V1, …, Vk connected by linear maps L1: V0 → V1, L2: V1 → V2, …, Lk: Vk−1 → Vk. The composite T = Lk ∘ … ∘ L1 is a linear map from V0 to Vk. A matrylinearny representation arranges these maps as blocks within a larger block matrix, so that the overall operator exhibits a multi-level, nested block structure. In practice, this means the action of T can be understood by inspecting both the individual linear components and their placement in a hierarchical matrix.

Examples include block matrices where each block itself represents a linear operator, producing a nested, multi-scale operator. Block-diagonal matrices with operator blocks, or matrices formed by Kronecker products that encode hierarchical relations, can be viewed as instances of matrylinearny structures. The concept is related to, but distinct from, standard block matrices and tensor-based representations used in multi-stage linear models.

Key properties often emphasized are linearity at every level and compatibility of composition across levels, which allows multi-level analysis without abandoning linear methods. Applications appear in numerical linear algebra, multi-resolution modeling, and structured representations in signal processing or certain neural network architectures.

Related concepts include block matrices, Kronecker products, and hierarchical linear models.