The Efficient Calculation of the Eigenvalues, Eigenvectors and Inverses of a Special Class of Brownian Matrices

A Brownian matrix is defined by having all the elements in any row, to the right of the principle diagonal, equal, and all elements in any column, below the principal diagonal, equal. These matrices occur in signal processing problems, and it has been shown that the solution of Brownian systems of linear equations can be solved in 0(n) flops. The inverse of a nonsingular Brownian matrix can be found in 0(n2) flops.In this paper we consider a special class of Brownian matrices, which form a ring, and show that both the normal inverse, when nonsingularity applies, and some generalized inverses can be found in 0(n) flops. In addition, explicit formulae for the determinant, the eigenvalues and eigenvectors are given, also requiring 0(n) flops.