[Paper Review] Compounding Doubly Affine Matrices

Adam Rogers, Ian Cameron+1 more|arXiv (Cornell University)|Nov 29, 2017

Matrix Theory and Algorithms7 references3 citations

TL;DR

This paper presents a general analytical method for constructing larger Integer Sequence Doubly Affine (ISDA) arrays—such as magic squares, Latin squares, and hypercubes—by compounding smaller ISDA matrices using weighted sums of left and right Kronecker products. The key contribution is deriving explicit formulas for the eigenvalues and singular values of the compound matrices, enabling exact rank computation and revealing spectral properties preserved across dimensions, with applications to high-order magic and Latin structures in arbitrary dimensions.

ABSTRACT

Weighted sums of left and right hand Kronecker products of Integer Sequence Doubly Affine (ISDA) as well as Generalized Arithmetic Progression Doubly Affine (GAPDA) arrays are used to generate larger ISDA arrays of multiplicative order (compound squares) from pairs of smaller ones. In two dimensions we find general expressions for the eigenvalues (EVs) and singular values (SVs) of the larger arrays in terms of the EVs and SVs of their constituent matrices, leading to a simple result for the rank of these highly singular compound matrices. Since the critical property of the smaller constituent matrices involves only identical row and column sums (often called semi-magic), the eigenvalue and singular value results can be applied to both magic squares and Latin squares. Additionally, the compounding process works in arbitrary dimensions due to the generality of the Kronecker product, providing a simple method to generate large order ISDA cubes and hypercubes. The first examples of compound magic squares are found in manuscripts that date back to the 10th century CE, and other representative applications are outlined through judicious examples.

Motivation & Objective

• To develop a general, analytically tractable method for constructing larger ISDA arrays from smaller seed matrices using Kronecker products.
• To derive closed-form expressions for eigenvalues and singular values of compound ISDA matrices in two dimensions.
• To extend the compounding framework to arbitrary dimensions, enabling construction of high-order magic cubes and hypercubes.
• To analyze the propagation of structural properties (e.g., diagonal sums, pandiagonality) under compounding, especially in Latin and magic arrays.
• To provide a unified spectral framework applicable to semi-magic, magic, and Latin squares through shared eigenvalue and singular value characteristics.

Proposed method

• The method uses weighted sums of left and right Kronecker products of smaller ISDA matrices to generate larger compound matrices of order $mn$ from seed matrices of order $n$ and $m$.
• Eigenvalues of the compound matrix are derived as products of eigenvalues from the constituent matrices, with a specific structure preserving the line sum as the dominant eigenvalue.
• Singular values of the compound matrix are expressed as scaled products of the singular values of the seed matrices, enabling exact computation of the full singular value spectrum.
• The rank of the compound matrix is analytically determined as $\text{rank}(\mathbf{C}) = \text{rank}(\mathbf{A}) + \text{rank}(\mathbf{B}) - 1$, validated through numerical examples.
• The approach generalizes to $D$-dimensional arrays using the Kronecker product's associative and distributive properties across multiple dimensions.
• The method is applied to construct compound magic squares (e.g., order 12), Latin cubes (e.g., order 8), and hypercubes, with explicit examples provided for validation.

Experimental results

Research questions

• RQ1 How can eigenvalues and singular values of compound ISDA matrices be expressed in terms of the spectral properties of their constituent matrices?
• RQ2 What is the exact rank of a compound ISDA matrix formed via Kronecker product compounding, and how does it relate to the ranks of the seed matrices?
• RQ3 Under what conditions are special properties like pandiagonality or diagonal uniformity preserved during the compounding process?
• RQ4 Can the compounding method be generalized to arbitrary dimensions to generate high-order magic cubes and hypercubes?
• RQ5 How do spectral invariants such as the $R$-index (sum of fourth powers of singular values) behave under compounding?

Key findings

• The eigenvalues of the compound matrix are formed as products of the eigenvalues of the seed matrices, with the dominant eigenvalue corresponding to the line sum, and complex conjugate pairs preserved.
• The singular values of the compound matrix are given by $\sigma_{i}^{2} = (a_i b_j)^2$ for scaled Kronecker products, with explicit expressions derived for the full spectrum.
• The rank of the compound matrix is exactly $\text{rank}(\mathbf{C}) = \text{rank}(\mathbf{A}) + \text{rank}(\mathbf{B}) - 1$, verified numerically for an order 12 magic square with rank 5.
• An order 12 magic square constructed from a Lo Shu square ($n=3$) and a $m=4$ magic square has eigenvalues $\lambda_1 = 858$, $\lambda_2 = 4(2i\sqrt{6})$, $\lambda_3 = -4(2i\sqrt{6})$, and singular values $\sigma_i^2 = \{858^2, 9^2 \times 320, 9^2 \times 20, 4^2 \times 48, 4^2 \times 12\}$.
• The compounding process successfully generates a valid order 8 Latin cube from two $N=2$ Latin cubes, preserving row, column, and pillar sums, with both aggregated and dispersed variants constructed.
• The method applies universally to ISDA arrays with $k=1$ (Latin), $k=2$ (magic squares), $k=3$ (magic cubes), and higher $k$, enabling construction of high-order hypercubes in arbitrary dimensions.