Matrices Whose Inversions are Tridiagonal, Band or Block-Tridiagonal and Their Relationship with the Covariance Matrices of a Random Markov Processes

Abstract

The article discusses the matrices of the forms A_n^1, A_n^m, A_N^m , whose inversions are: tridiagonal matrix A_n^−1 (n - dimension of the A_N^-m
N matrix), banded matrix A_n^−m (m is the half-width band of the matrix)
or block-tridiagonal matrix A_N^−m (N = n x m – full 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 frequently occur in the problems of optimal filtering, extrapolation and interpolation of MRP and Markov Random Fields (MRF). It is shown, that the structures of the matrices A_n^1, A_n^m, A_N^m have the same form, but the matrix elements in the first case are scalar quantities; in the second case matrix elements represent 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 were found. Also computational efficiency in the storage and inverse of suchmatrices have been considered. To illustrate the acquired results, an example on the covariance matrix inversions of two-dimensional MRP is given.