aij2ij

aij2ij is a compact, informal notation sometimes used in mathematical or computational literature to denote a transformation that takes a two-index object aij (such as a matrix A with entries aij) and maps it to another two-index object with indices that may be relabeled or permuted. It is not a universally standardized operator, but a shorthand for describing a class of index-mapping operations.

Formally, let A be a matrix or rank-2 tensor with entries aij, where i and j range over finite index sets I and J. The operation denoted by aij2ij describes a mapping T that produces B with entries bij′ = aπ(i′)ρ(j′), where π: I → I and ρ: J → J are bijections (index relabelings). In this framing, aij2ij encompasses a family of transformations including simple relabeling, row/column permutations, and more complex index rearrangements. A common special case is the transposition, where the indices are swapped: Bij = aji.

Examples illustrate the idea. Transposition is the familiar case where aij2ij yields a matrix B with Bij = Aji. A more general example is applying fixed permutations to the row and column indices, so that Bij = Aπ(i)ρ(j), effectively reordering rows by π and columns by ρ without changing the underlying data values, only their positions.

Usage and note. In texts that employ aij2ij, the exact meaning depends on the author’s convention, so the specific π and ρ, or the direction of the mapping, should be stated. The notation is often used to succinctly describe index-based reshaping, reordering, or permutation operations in linear algebra, tensor calculus, and related computational algorithms.