Analytical Inverse For The Symmetric Circulant Tridiagonal Matrix

TL;DR

This paper introduces an exact, efficient analytical method for inverting symmetric circulant tridiagonal matrices and solving related linear systems, which are common in engineering and applied mathematics.

Contribution

It presents a novel decomposition approach for symmetric circulant matrices, enabling stable and exact inversion and solution of linear systems with reduced computational complexity.

Findings

Exact inverse formulas for symmetric circulant matrices derived

Decomposition method enables efficient linear system solving

Method maintains stability comparable to other direct methods

Abstract

Finding the inverse of a matrix is an open problem especially when it comes to engineering problems due to their complexity and running time (cost) of matrix inversion algorithms. An optimum strategy to invert a matrix is, first, to reduce the matrix to a simple form, only then beginning a mathematical procedure. For symmetric matrices, the preferred simple form is tridiagonal. This makes tridiagonal matrices of high interest in applied mathematics and engineering problems. This study presents a time efficient, exact analytical approach for finding the inverse, decomposition, and solving linear systems of equations where symmetric circulant matrix appears. This matrix appears in many researches and it is different from ordinary tridiagonal matrices as there are two corner elements. For finding the inverse matrix, a set of matrices are introduced that any symmetric circulant matrix could be decomposed into them. After that, the exact analytical inverse of this set is found which gives the inverse of circulant matrix. Moreover, solving linear equations can be carried out using implemented decomposition where this matrix appears as a coefficient matrix. The methods principal strength is that it is as stable as any other direct methods (i.e., execute in a predictable number of operations). It is straightforward, understandable, solid as a rock and an exceptionally good psychological backup for those times when something is going wrong and you think it might be your linear equation solver. The downside of present method, and every other direct method, is the accumulation of round off errors.