Negative integral powers of a bidiagonal matrix

Abstract:

The elements of the inverse of a bidiagonal matrix have been expressed in a convenient form. The higher negative integral powers of the bidiagonal matrix exhibit an interesting property: the (ij)th element of the $( - m)$th power is equal to the product of the corresponding element of the inverse by a Wronski polynomial, viz., the complete symmetric function of degree $(m - 1)$ of the diagonal elements, ${d_i},{d_{i + 1}}, \ldots ,{d_j}$, of the inverse matrix.