On triangular similarity of nilpotent triangular matrices

Abstract

Let Bn (resp. Un, Nn) be the set of n × n nonsingular (resp. unit, nilpotent) upper triangular matrices. We use a novel approach to explore the Bn-similarity orbits in Nn. The Belitskiȋ's canonical form of A ∈ Nn under Bn-similarity is in QUn where Q is the subpermutation such that A ∈ BnQBn. Using graph representations and Un-similarity actions stabilzing QUn, we obtain new properties of the Belitskiȋ's canonical forms and present an efficient algorithm to find the Belitskiȋ's canonical forms in Nn. As consequences, we construct new Belitskiȋ's canonical forms in all Nn's, list all Belitskiȋ's canonical forms for n = 7, 8, and show examples of 3-nilpotent Belitskiȋ's canonical forms in Nn with arbitrary numbers of parameters up to O(n 2 ).Mathematics Subject Classification 2020: Primary 15A21, Secondary 15A23, 14L30.