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

TL;DR

This paper explores matrices with special inverse structures related to Markov processes, providing formulas for their inversion, analyzing their properties, and demonstrating their application to covariance matrices in stochastic processes.

Contribution

It introduces a unified approach to invert matrices with specific structures linked to Markov processes and derives simple formulas for their inverses.

Findings

Derived formulas for inverses of structured matrices

Analyzed properties and computational efficiency of these matrices

Provided an example with two-dimensional Markov Random Processes

Abstract

The article discusses the matrices of the three forms whose inversions are: tridiagonal matrix, banded matrix or block-tridiagonal matrix and their relationships with the covariance matrices of measurements of 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 these three forms of matrices 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.