Echelon Type Canonical Forms in Upper Triangular Matrix Algebras

It is common knowledge that matrices can be brought in echelon form by Gaussian elimination and that the reduced echelon form of a matrix is canonical (also) in the sense that it is unique. A crucial auxiliary result in [BW] suggests a generalization of the standard echelon form. For square matrices, some new canonical forms of echelon type are introduced. One of them (suggested by observations made in [Lay] and [SW]) has the important property of being an upper triangular idempotent. The others come up when working exclusively in the context of \( \mathbb{C}^{n \times n}_{upper} \), the algebra of upper triangular n × n matrices. Subalgebras of \( \mathbb{C}^{n \times n}_{upper} \) determined by a pattern of zeros are considered too. The issue there is whether or not the canonical forms referred to above belong to the subalgebras in question. In general they do not, but affirmative answers are obtained under certain conditions on the given preorder which allow for a large class of examples and that also came up in [BES4]. Similar results hold for canonical generalized diagonal forms involving matrices for which all columns and rows contain at most one nonzero entry. The new canonical forms are used to study left, right and left/right equivalence in zero pattern algebras. For the archetypical full upper triangular case a connection with the Stirling numbers (of the second kind) and with the Bell numbers is made.