Rational G-Circulants Satisfying the Matrix Equation A² = dI + λJ

Citation

Lam, Clement Wing Hong (1974) Rational G-Circulants Satisfying the Matrix Equation A² = dI + λJ. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/7qh9-nn17. https://resolver.caltech.edu/CaltechTHESIS:02022021-194446003

Abstract

A g-circulant is a square matrix of rational numbers in which each row is obtained from the preceding row by shifting the elements cyclically g columns to the right. This work studies g-circulants A which satisfy the matrix equation A² = dI + λJ, where I is the identity matrix and J is the matrix of 1's. Necessary and sufficient conditions are given for the existence of solutions when g = 1. The existence of (0,1) g-circulants satisfying A² = dI + λJ is shown to be equivalent to the existence of (v, k, λ, g)-addition sets, which are generalizations of difference sets. It is proved that there are no nontrivial (v, k, λ, 1)-addition sets. Some examples of (v, k, λ, g)-addition sets are given and the multiplier theorem for (v, k, λ, g)-addition sets is also proved.

Thesis Files