A Note on Idempotent Matrices: The Poset Structure and The Construction
Sen-Peng Eu, Yong-Siang Lin, Wei-Liang Sun
TL;DR
This work analyzes idempotent matrices from two angles: a poset-theoretic viewpoint over division rings and a constructive approach over UFDs and PIDs. It yields a precise block-decomposition criterion for the partial order on idempotents in Mn(Δ) via $F = A (I_{rank(E)}\;0;0\;T) A^{-1}$, with $T$ idempotent, and shows that intervals in the poset correspond to smaller-idempotent structures. In the PID setting it leverages Smith normal form to obtain explicit parametrizations of idempotents, including a concrete form $E = (C A C B D)$ with $AC+BD=I_{\ell}$, and extends to constructions using Kronecker products and anti-transpose. Finally, it views idempotents over a field as an affine algebraic variety, establishing dimensional bounds and irreducibility properties of the rank strata. The results clarify both the structural and constructive aspects of idempotents in matrix rings and link them to geometric viewpoints via varieties.
Abstract
Idempotent elements play a fundamental role in ring theory, as they encode significant information about the underlying algebraic structure. In this paper, we study idempotent matrices from two perspectives. First, we analyze the partially ordered set of idempotents in matrix rings over a division ring. We characterize the partial order relation explicitly in terms of block decompositions of idempotent matrices. Second, over principal ideal domains, we establish an equivalent condition for a matrix to be idempotent, derived from matrix factorizations using the Smith normal form. We also consider extensions over unique factorization domains and constructions via the Kronecker product and the anti-transpose. Together, these results clarify both the structural and constructive aspects of idempotents in matrix rings. Moreover, the set of idempotent matrices over a field can be viewed as an affine algebraic variety.