On the Inverses of Brownian and Brownian-Like Matrices

In several papers by F. Valvi, sufficient conditions are given for Brownian and Brownian-like matrices to have Hessenberg inverses. We interpret these conditions from the viewpoint of familiar facts related to matrices of small triangular rank. This allows us to formulate more general assertions on the Hessenberg property of the inverse. Moreover, we explicitly find the structure of the inverse of a Brownian matrix under a certain natural irreducibility condition. This structure is similar to the well-known structure of the inverse of an irreducible tridiagonal matrix. Furthermore, we show that the parameters defining the inverse of an ($n\times n$) Brownian matrix can be calculated in $O(n)$ arithmetic operations.