Factorization of banded permutations

Abstract

Abstract. We consider the factorization of permutations into bandwidth 1 permutations, which are products of mutually nonadjacent simple transpositions. We exhibit an upper bound on the minimal number of such factors and thus prove a conjecture of Gilbert Strang: a banded permutation of bandwidth w can be represented as the product of at most 2w − 1 permutations of bandwidth 1. An analogous result holds also for infinite and cyclically banded permutations.

Groups of banded matrices with banded inverses

“…We conjectured that fewer than 2w factors F would be sufficient. Greta Panova has found a beautiful proof [6]. Other constructions [1], [9] also yield N ≤ 2w − 1.…”

“…At each step we move from left to right, exchanging pairs of neighbors that are in the wrong order: We conjectured in [10] that N ≤ 2w −1 in all cases. A beautiful proof is given by Greta Panova [6], using the "wiring diagram" to decide the sequence of exchanges in advance. A second proof [1] by Albert, Li, Strang, and Yu confirms that the algorithm illustrated above also has N ≤ 2w − 1.…”

The main diagonal of a permutation matrix

“…The centered matrix A c = S κ A has index + (A c ) = 0 (see (16) below). This is necessary for a triangular matrix to have a triangular inverse of the same kind (Corollary 6.4).…”

“…The intersections of those lines tell us the order in which to exchange neighbors. Key point: All intersections lie on N < 2w vertical lines in the [16,23]. Exchanges on each vertical line can be executed in parallel by a matrix F i (its bandwidth is 1).…”

Permutations as Product of Parallel Transpositions

“…Panova [2] has found a particularly elegant proof based on wiring diagrams. Her approach shows all factors at once, where our construction reaches the identity permutation by a sequence of parallel exchanges of neighbors (which are permutations of bandwidth 1).…”