Algebra

Matrix Theoretic and Combinatorial Structures in Addition Chains

Authors: Theophilus Agama
Comments: 23 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

An addition chain of length $h$ that leads to a number $n$ is a sequence of positive integers $s_0=1,s_1=2,ldots,s_h=n$ such that $s_i=s_j+s_k$~($i>jgeq k$) for each $1leq ileq h$. We introduce a matrix-theoretic framework for studying the properties of arbitrary addition chains that lead to a target integer $ngeq 2$ by associating each chain and the sequence of dominant and lower weight summands that calculate each term in the chain an adjacency matrix that encodes the internal geometry. In this linear-algebraic and matrix theoretic framework, we investigate how rank, rank profiles, singular values, eigenvalues, matrix norms, predecessor depths, and track-interaction terms encode the combinatorial and arithmetic structure of addition chains.
Category: Algebra