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

“Let T_n = tridiag(-1,b,-1), an n x n symmetric, strictly diagonally dominant tridiagonal matrix (|b| > 2). This paper investigates tridiagonal near-Toeplitz matrices \widetilde{T}n := [\wtt{i,j}], obtained by perturbing the (1,1) and (n,n) entry of T_n. Let \wtt_{1,1} = \wtt_{n,n} = \wtb \neq b. We derive exact inverses of \wtT_n. Furthermore, we demonstrate that these results hold even when |\widetilde{b}| < 1. Additionally, we establish upper bounds for the infinite norms of the inverse matrices.

The row sums and traces of the inverse provide insight into the matrix’s spectral properties and play a key role in understanding the convergence of fixed-point iterations. These metrics allow us to derive tighter bounds on the infinite norms and improve computational efficiency.

Numerical results for Fisher’s problem demonstrate that the derived bounds closely match the actual infinite norms, particularly for b > 2 with \widetilde{b} \leq 1 and b < -2 with \widetilde{b} \geq -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.”