Products and inverses of multidiagonal matrices with equally spaced diagonals

TL;DR

This paper studies the algebraic properties of multidiagonal matrices with equally spaced diagonals, proving closure under multiplication, powers, and inverses, and providing explicit formulas for inverses in certain cases.

Contribution

It establishes that a class of multidiagonal matrices with equally spaced diagonals is closed under key algebraic operations and derives explicit inverse formulas for specific cases.

Findings

The set of such multidiagonal matrices is closed under multiplication and positive powers.

Nonsingular matrices in this class are closed under taking inverses and negative powers.

Explicit inverse formulas are provided for certain multidiagonal matrices.

Abstract

Let $n,k$ be fixed natural numbers with $1\le k\le n$ and let $A_{n+1,k,2k,\dots,sk}$ denote an $(n+1)\times (n+1)$ complex multidiagonal matrix having $s=[n/k]$ sub- and superdiagonals at distances $k,2k,\dots,sk$ from the main diagonal. We prove that the set $\mathcal{MD}_{n,k}$ of all such multidiagonal matrices is closed under multiplication and powers with positive exponents. Moreover the subset of $\mathcal{MD}_{n,k}$ 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 $\mathcal{MD}_{n,k}$, moreover if $n+1\le 2k$ then $A^{-1}_{n+1,k}$ is also $k$-tridigonal. Using this fact we give an explicite formula for this inverse.