Matricesthose

Matricesthose is a hypothetical construct in linear algebra and data science used to describe a family of matrices generated by applying a selection criterion to rows, columns, or both. It is not a standard term in widely used mathematical literature, but it appears in discussions of submatrix families and masked computations.

Definition: Let A be an m-by-n matrix. A matricesthose associated with a selection rule S consists of a collection {A[S(i)]} for i in an index set I, where S(i) specifies a subset of row indices, column indices, or both. The resulting submatrices are obtained by taking the intersection of the indicated rows and columns. The rule S may be predicated on the entries of A or on external criteria, yielding a family parameterized by I.

Properties: The family is closed under intersection of selections: if i and j yield S(i) and S(j), their intersection S(i) ∩ S(j) yields a submatrix contained in both. If S is monotone with respect to a partial order on I, the corresponding submatrices are ordered by inclusion. In the special case where S(i) selects a boolean mask, matricesthose reduces to a set of masked submatrices.

Examples: Example 1: A 4x4 matrix with S(1) selecting rows {1,3}, columns {2,4}; Example 2: a pattern where S(i) chooses rows based on a threshold applied to row sums, yielding different submatrices.

Applications: The concept is used to illustrate submatrix families, multi-view data representations, and strategies for localized computations in algorithms.

Etymology and usage: The term blends "matrix" with the pronoun "those" to emphasize a family of substructures. It has not gained widespread traction and is mainly found in speculative or didactic contexts.

See also: matrix, submatrix, masking, boolean mask, indexing.