Application of quasideterminants to the inverse of block triangular matrices over noncommutative rings

TL;DR

This paper introduces new explicit formulas for inverting block triangular matrices over noncommutative rings using quasideterminants, enabling perturbation analysis and practical computation.

Contribution

It provides two novel representations of the inverse of block triangular matrices over noncommutative rings, one using quasideterminants and the other via recurrence relations.

Findings

Explicit inverse formulas using quasideterminants.

Recurrence-based representation for inverse blocks.

Effective perturbation analysis demonstrated with an example.

Abstract

Given a block triangular matrix $M$ over a noncommutative ring with invertible diagonal blocks, this work gives two new representations of its inverse $M^{-1}$. Each block element of $M^{-1}$ is explicitly expressed via a quasideterminant of a submatrix of $M$ with the block Hessenberg type. Accordingly another representation for each inverse block is attained, which is in terms of recurrence relationship with multiple terms among blocks of $M^{-1}$. The latter result allows us to perform an off-diagonal rectangular perturbation analysis for the inverse calculation of $M$. An example is given to illustrate the effectiveness of our results.