Explicit Inversion for Two Brownian-Type Matrices

TL;DR

This work derives explicit inverses and determinant formulas for two Brownian-type matrices constructed as $A_1 = K \circ G_n$ and $A_2 = N \circ G_n$, parameterized by $3n-1$ values. The inverses are shown to be lower Hessenberg with closed-form entries, obtained via a constructive row-operation sequence, and matching determinant expressions reveal non-singularity conditions. The authors analyze computational cost, presenting a recurrence-based algorithm that computes $A_1^{-1}$ (and $A_2^{-1}$) in $O(n^2)$ operations, significantly outperforming LU decomposition for large $n$. The results generalize classical test matrices and provide efficient tools for matrix inversion testing in digital signal processing contexts.

Abstract

We present explicit inverses of two Brownian--type matrices, which are defined as Hadamard products of certain already known matrices. The matrices under consideration are defined by $3n-1$ parameters and their lower Hessenberg form inverses are expressed analytically in terms of these parameters. Such matrices are useful in the theory of digital signal processing and in testing matrix inversion algorithms.