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

2020/06/28 by Zhan, Xuzhou
#15A09 #15A15 #15B33 #47A05 #47A55 #65F05 #FOS: Mathematics #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2006.15544

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.

Related

• Elementary triangular matrices and inverses of k-Hessenberg and triangular matrices
• A New Algorithm for the Inverse of Matrices with Noncommuting Entries
• Algorithms for Recursive Block Matrices
• Quasideterminants
• Noncommutative Spectral Decomposition with Quasideterminant