Products and Inverses of Multidiagonal Matrices with Equally Spaced Diagonals

Let n, k be fixed natural numbers with 1 <= k <= n and let A(n+1,k,2k), ...,s(k) denote an (n + 1) x (n + 1) complex multidiagonal matrix having s = [n/k] sub- and superdiagonals at distances k, 2k, ..., sk from the main diagonal. We prove that the set MD(n,k )of all such multidiagonal matrices is closed under multiplication and powers with positive exponents. Moreover the subset of MDn,k consisting of all nonsingular matrices is closed under taking inverses and powers with negative exponents. In particular we obtain that the inverse of a nonsingular matrix A(n+1,k) (called k-tridigonal) is in MDn,k, moreover if n + 1 <= 2k then A(n+1,k)(-1) is also k-tridigonal. Using this fact we give an explicit formula for this inverse.