arccostraceR

arccostraceR is a hypothetical statistical operator and an accompanying R package designed to compute the arccosine of the normalized trace of a square matrix. The central function, also named arccostraceR, takes a numeric matrix and returns a scalar or a vector of scalars depending on the input’s shape.

Mathematical background and definition: for a square matrix A with dimension n, the default operation is to compute t = trace(A) / n, where trace(A) is the sum of the diagonal elements. The function then returns arccos(t), with the result expressed in radians. This construction ties the output to the mean eigenvalue of A, since trace(A)/n equals the average of A’s eigenvalues. The arccos argument must lie in the interval [-1, 1]; if trace(A)/n falls outside this domain, arccostraceR issues a warning and returns a missing value for that input.

Usage and input handling: arccostraceR accepts a single square matrix or a list (or array) of matrices and applies the operation elementwise. By default, normalization by n is performed, but the function can be extended to alternative normalization schemes in user implementations. The behavior is defined to be deterministic and suitable for exploratory analysis or teaching contexts.

Example and interpretation: for the identity matrix I_n, trace(I_n) = n, so trace(I_n)/n = 1 and arccos(1) = 0. For matrices with varied diagonal values, the result reflects how far the mean eigenvalue deviates from 1. The metric is primarily of instructional value, illustrating how a matrix statistic maps through a nonlinear transform to a scalar.

Notes: arccostraceR is a conceptual rather than a widely adopted real-world tool. It serves as an illustrative example of how matrix-derived statistics can be transformed and interpreted in a compact, scalar form. See also arccos, trace, eigenvalues, and matrix normalization.

---