Fast multiplication, determinants, and inverses of arrowhead and diagonal-plus-rank-one matrices over associative fields

TL;DR

This paper presents unified, efficient formulas for fast multiplication, determinants, and inverses of arrowhead and diagonal-plus-rank-one matrices over various associative fields, applicable in both commutative and noncommutative cases.

Contribution

It introduces unified formulas that work across different algebraic fields, including noncommutative quaternions, for key matrix operations with optimal computational complexity.

Findings

Formulas require O(n) operations for matrix-vector multiplication.

Determinants and inverses are computed efficiently with unified formulas.

Applicable to block matrices and both commutative and noncommutative fields.

Abstract

The article considers arrowhead and diagonal-plus-rank-one matrices in F^(nxn) where F in R,C or H. H is a non-commutative field of quaternions. We give unified formulas for fast matrix-vector multiplications, determinants, and inverses for considered matrices. The formulas are unified in the sense that the same formula holds in both, commutative and noncommutative algebras. Each formula requires O(n) arithmetic operations. Most of the formulas hold for block matrices, as well.