Compound Lucas Magic Squares

TL;DR

This paper explores the mathematical properties and parameterizations of compound Lucas magic squares, including their matrix decompositions, naturalness criteria, and relationships to other parameterizations, extending understanding of magic square structures.

Contribution

It introduces a comprehensive parameterization of compound Lucas magic squares, analyzes their matrix forms, and develops criteria for their naturalness, expanding the theoretical framework of magic squares.

Findings

Derived matrix expressions in Jordan canonical form and SVD.

Established criteria for natural compound Lucas magic squares.

Connected Lucas parameterization to Frierson's approach.

Abstract

We review a general parameterization of an order-3 magic square derived by Lucas and we compound it to produce a parameterized order-9 magic square. Sequential compounding to higher order also is treated. Expressions are found for the matrices in the Jordan canonical form and the singular value decomposition of the compound Lucas magic square matrices. We develop a procedure for determining if an order-n magic square may be natural. This enables determination of numerical values for parameters in natural compound Lucas magic squares. Also, we find commuting pairs of compound Lucas matrices and formulas for matrix powers of order-3 and order-9 Lucas matrices. A parameterization due to Frierson is related to Lucas' parameterization and our results specialize to it, complementing previous results.