Similarity for zero-square matrices

TL;DR

This paper investigates the similarity classes of zero-square matrices over various rings, establishing conditions under which such matrices are similar to scalar multiples of a specific matrix, and identifying cases where this is not possible.

Contribution

It characterizes when zero-square matrices are similar to multiples of E_1n over GCD domains and shows limitations for larger matrices.

Findings

Matrices over GCD domains with n=2 are similar to multiples of E_1n.

For n=3, similar matrices exist if 2 is not a zero divisor.

For n≥4, not all matrices are similar to such multiples over any ring.

Abstract

Let T be an n by n zero-square matrix over a commutative unital ring R. We show that T is similar to a multiple of E_1n if R is a GCD domain and n = 2, if R is a GCD domain with 2 not zero divisor and n = 3, but there are matrices which are not similar to any multiple of E_1n whenever n greater or equal 4, over any commutative unital ring.