Some fast algorithms multiplying a matrix by its adjoint

TL;DR

This paper introduces a new non-commutative algorithm for multiplying matrices by their adjoint, reducing complexity, and provides specialized algorithms for complex numbers and quaternions, with practical implementation strategies.

Contribution

It presents a novel algorithm for matrix-adjoint multiplication that improves efficiency and offers specialized methods for complex and quaternion matrices, along with practical implementation schedules.

Findings

Reduced matrix-adjoint multiplication to general matrix product with constant factor improvement

Proved no 4-product bilinear form algorithm exists for this problem

Developed low-memory schedules for efficient implementation over prime fields

Abstract

We present a non-commutative algorithm for the multiplication of a 2 x 2 block-matrix by its adjoint, defined by a matrix ring anti-homomorphism. This algorithm uses 5 block products (3 recursive calls and 2 general products)over C or in positive characteristic. The resulting algorithm for arbitrary dimensions is a reduction of multiplication of a matrix by its adjoint to general matrix product, improving by a constant factor previously known reductions. We prove also that there is no algorithm derived from bilinear forms using only four products and the adjoint of one of them. Second we give novel dedicated algorithms for the complex field and the quaternions to alternatively compute the multiplication taking advantage of the structure of the matrix-polynomial arithmetic involved. We then analyze the respective ranges of predominance of the two strategies. Finally we propose schedules with low memory footprint that support a fast and memory efficient practical implementation over a prime field.