INVERSE OF A CERTAIN BAND TOEPLITZ MATRIX

摘要

In his paper 'Inversion of certain symmetric band matrices', Lars Rehnqvist gives an algorithm for the inverse of a band Toeplitz matrix A of order nxn arising from certain statistical problems. The elements of A are a_(i, j) = k - |i - j|, if |i - j| < k and a_(i,j) = 0, if |i - j| ≥ k for integer k ≤ n. In this paper a key idea of Rehnqvist is exploited to find the exact inverse of a generalization of A. The result confirmed Rehnqvist report that the inverse matrix is dependant on the value of k and n. The determinant of the matrix is also found and thereby proving a conjecture by E.L. Allgower. A range of values for x that guarantees the non-singularity of the matrix is also determined.