Matrices whose inverses are tridiagonal, band or block-tridiagonal and their relationship with the covariance matrices of a random Markov process

Abstract

The article discusses the matrices of the 1 n A , m n A , m N A forms, whose inversions are: tridiagonal matrix 1  n A (ndimension of the matrix), banded matrix m n A  (m -the half-width band of the matrix) or block-tridiagonal matrix m N A  (N=n x mfull dimension of the block matrix; m -the dimension of the blocks) and their relationships with the covariance matrices of measurements with ordinary (simple) Markov Random Processes (MRP), multiconnected MRP and vector MRP respectively. Such covariance matrices are frequently occurring in the problems of optimal filtering, extrapolation and interpolation of MRP and Markov Random Fields (MRF). It is shown, that the structure of the matrix 1 n A , m n A , m N A , has the same form, but the matrix elements in the first case are scalar quantities; in the second case matrix elements representing a product of vectors of dimension m; and in the third case, the off-diagonal elements are the product of matrices and vectors of dimension m. The properties of such matrices were investigated and a simple formulas of their inversion was founded. Also computational efficiency in the storage and inverse of such matrices have been considered. To illustrate the acquired results an example of the covariance matrix inversions of two-dimensional MRP is given. Index Terms-Best Linear Unbiased Estimates (BLUE), Markov process in the wide sense, simple (ordinary connected) Markov process, multiply connected (m-connected) Markov process, vector (m-dimensional) Markov process, Random field filtering and parametric identification, Tridiagonal Matrices, Banded Matrices and Block-Tridiagonal matrices.

Determinants of some pentadiagonal matrices

“…The research on multidiagonal, in particular tridiagonal and pentadiagonal matrices, intensified in the past years. These matrices have important applications in optimization problems [3], autoregression modelling [26], approximation theory [23], Gauss-Markov random processes [2], orthogonal polynomials, solving elliptic and parabolic PDE's with finite difference methods [10], inequalities (quadratic, Wirtinger, Opial's type) [4], [18].…”