arctandiagonal

Arctandiagonal is a term encountered in some mathematical discussions as a contraction of arctan and diagonal. There is no widely accepted, formal definition, and usage varies by author or context. In practice, the phrase is often used to describe ideas surrounding how the arctangent function interacts with diagonal structures in matrices or grids.

In linear algebra, a common interpretation centers on functional calculus. If A is diagonalizable as A = P D P^{-1}, then applying arctan via a matrix function yields arctan(A) = P arctan(D) P^{-1}. In this setting, the diagonal form of arctan(A) in the eigenbasis has diagonal entries arctan(lambda_i), where lambda_i are the eigenvalues of A. When A is itself diagonal, arctan acts entrywise, producing a new diagonal matrix whose diagonal entries are arctan(lambda_i). In these contexts, the term arctandiagonal is sometimes used to indicate the diagonal-like outcome after applying the arctan transformation.

In other usages, arctandiagonal may refer more loosely to diagonal patterns that arise from taking arctangent-based transformations of coordinate data, or to geometric constructions where diagonal lines or vectors are defined by arctangent relationships. Because the term is not standardized, different fields may emphasize different aspects, such as diagonalization behavior under the arctan function or the appearance of arctan-derived diagonals in grids and representations.

See also: main diagonal, anti-diagonal, diagonalization, matrix function, arctan.