A Note on Idempotent Matrices: The Poset Structure and The Construction (2510.09501v1)

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.

• The paper demonstrates that idempotent matrices follow a partial order defined by E ≤ F when EF = E = FE, clarified through block decompositions.
• It employs methodologies such as the Smith normal form and Kronecker product to construct idempotent matrices over principal ideal domains.
• The study’s findings offer structural insights that could enhance algorithmic applications in symbolic computation and systems theory.

Analyzing "A Note on Idempotent Matrices: The Poset Structure and The Construction"

The paper "A Note on Idempotent Matrices: The Poset Structure and The Construction" explores the structural properties and constructive methodologies of idempotent matrices over different algebraic structures. The discussion is twofold: the partial order of idempotents in matrix rings over division rings and the construction of idempotent matrices using tools such as the Smith normal form, especially for principal ideal domains (PIDs).

Poset Structure of Idempotents

The paper introduces the concept of a partial order on idempotent matrices where $E \le F$ if $EF = E = FE$. This partial order plays a crucial role in understanding the algebraic structure of these matrices:

• Block Decomposition: Idempotent matrices in matrix rings over division rings can be characterized by their block decompositions. This allows a clear criterion for comparing them under the defined partial order.
• Diagonalization in Division Rings: An idempotent matrix $E$ over a division ring can be expressed as $E = ADA^{-1}$ where $D$ is a block diagonal matrix containing identity and zero matrices.

Constructive Methods for Idempotents

The paper provides methods to construct idempotent matrices over PIDs, using matrix factorizations:

• Smith Normal Form: Over PIDs, the matrix is decomposed to aid in identifying idempotent matrices. From this decomposition, conditions for a matrix to be idempotent are derived.
• Kronecker Product and Anti-transpose: Additional constructions involve the Kronecker product and operations like anti-transposing a matrix. These constructions illuminate alternative views of creating idempotent matrices.

Implications and Applications

Understanding idempotent matrices and their partial order has significant theoretical implications in ring theory and practical applications in linear algebra and computational mathematics. By identifying the structured way these matrices interact, the paper opens pathways for efficient algorithms to work with matrix rings, especially in symbolic computation and systems theory. Future research may explore algorithmic implementations for automated reasoning in algebraic systems, using the poset structure as a framework for optimization and partitioning tasks.