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.
翻译:我们考虑将带宽1变换的因子化为带宽1变换,这是相互不相邻的简单变换的产物。 我们展示了此类因素最少数量的上限,从而证明了吉尔伯特·斯特朗的推测:带宽的带宽变换可以代表带宽最多为2w-1美元的乘差的产物。 类似的结果对于无限和周期性的带宽变换也有影响。