Explicit inverse of symmetric, tridiagonal near Toeplitz matrices with strictly diagonally dominant Toeplitz part

Abstract

Let Tn=tridiag(−1,b,−1){T}_{n}={\rm{tridiag}}\left(-1,b,-1), an n×nn\times n symmetric, strictly diagonally dominant tridiagonal matrix (∣b∣>2| b| \gt 2). This article investigates tridiagonal near-Toeplitz matrices T˜n≔[t˜i,j]{\widetilde{T}}_{n}:= \left[{\widetilde{t}}_{i,j}], obtained by perturbing the (1,1)\left(1,1) and (n,n)\left(n,n) entry of Tn{T}_{n}. Let t˜1,1=t˜n,n=b˜≠b{\widetilde{t}}_{1,1}={\widetilde{t}}_{n,n}=\widetilde{b}\ne b. We derive exact inverses of T˜n{\widetilde{T}}_{n}. Furthermore, we demonstrate that these results hold even when ∣b˜∣2b\gt 2 with b˜≤1\widetilde{b}\le 1 and b<−2b\lt -2 with b˜≥−1\widetilde{b}\ge -1. For other cases, further refinement of the bounds is possible. Our results contribute to improving the convergence rates of fixed-point iterations and reducing the computation time for matrix inversion.

Keywords

• toeplitz matrices
• diagonal dominance
• exact inverses
• upper bounds
• 15a60
• 15b05
• 34l15