Inverses of k-Toeplitz matrices with applications to resonator arrays with multiple receivers

Abstract

We find closed-form algebraic formulas for the elements of the inverses of tridiagonal 2- and 3-Toeplitz matrices which are symmetric and have constant upper and lower diagonals. These matrices appear, respectively, as the impedance matrices of resonator arrays in which a receiver is placed over every 2 or 3 resonators. Consequently, our formulas allow to compute the currents of a wireless power transfer system in closed form, allowing for a simple, exact and symbolic analysis thereof. Small numbers are chosen for illustrative purposes, but the elementary linear algebra techniques used can be extended to k -Toeplitz matrices of this special form with k arbitrary, hence resonator arrays with a receiver placed over every k resonators can be analysed in the same way.