×

The Hadamard product of two Brownian matrices: Analytic inverse and determinant. (English) Zbl 0836.65032

A fast algorithm is developed for computing the explicit inversion \(B^{-1} = [\sigma_{i,j}]\) and determinant of \(B = N \circ A\), where \(\circ\) denotes the Hadamard product and \(N\), \(A\) are Brownian matrices. The inverse \(B^{-1}\) is found to be an upper Hessenberg matrix, the formulae of its elements being given analytically. A recursion relation provides the recursive algorithm for the elements of \(B^{-1}\). The recursive formula gives the inverse in \(O(n^2)\) multiplications/divisions and \(O(n)\) additions/subtractions. The elements of \(\text{det} (B)\) are also derived. The results may be of interest in digital signal processing.
Reviewer: V.Burjan (Praha)

MSC:

65F05 Direct numerical methods for linear systems and matrix inversion
65F40 Numerical computation of determinants
15A15 Determinants, permanents, traces, other special matrix functions
15A09 Theory of matrix inversion and generalized inverses
15B57 Hermitian, skew-Hermitian, and related matrices
PDFBibTeX XMLCite
Full Text: DOI