arXiv subjects

Ronald P. Nordgren

Publications and source records attributed to Ronald P. Nordgren.

5 recordsLinked to original sources

arXiv2021

Compound Lucas Magic Squares

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.

math.GM↗

arXiv2020

Commuting Magic Square Matrices

We review a known method of compounding two magic square matrices of order m and n with the all-ones matrix to form two magic square matrices of order mn. We show that these compounded matrices commute. Simple formulas are derived for their Jordan form and singular value decomposition. We verify that regular (associative) and pandiagonal commuting magic squares can be constructed by compounding. In a special case the compounded matrices are similar. Generalization of compounding to a wider class of commuting magic squares is considered. Three numerical examples illustrate our theoretical results.

math.GM↗

arXiv2020

Simultaneous Diagonalization and SVD of Commuting Matrices

We present a matrix version of a known method of constructing common eigenvectors of two diagonalizable commuting matrices, thus enabling their simultaneous diagonalization. The matrices may have simple eigenvalues of multiplicity greater than one. The singular value decomposition (SVD) of a class of commuting matrices also is treated. The effect of row/column permutation is examined. Examples are given.

math.GM↗

arXiv2018

Eulerian Composition of Certain Franklin Squares

Several specific Franklin squares and magic squares are decomposed into their quotient and remainder squares. The results support the conjecture that Franklin used the Eulerian composition method to construct many of his squares. This method also can be used to construct new Franklin squares as illustrated herein.

math.NT↗

arXiv2013

Pandiagonal and Knut Vik Sudoku Squares

We present systematic methods of constructing pandiagonal sudoku squares of order k*k and Knut Vik sudoku squares of order k*k not divisible by 2 or 3. Pandiagonal magic squares are constructed from these squares. Examples of all these squares are given.

math.HO↗