Orthogonal tripotent matrices

TL;DR

This work addresses the problem of characterizing orthogonal tripotent matrices, i.e., matrices $A$ satisfying $A^3=A=A^*$, and relates them to a broad landscape of matrix classes. The authors employ the Hartwig-Spindelböck decomposition, mean equations involving $A$, $A^*$, and $A^\dagger$, and powers of $AA^*$ and $A^*A$, supplemented by singular value decomposition criteria to derive a comprehensive set of necessary and sufficient conditions. Key contributions include multiple equivalent characterizations of $\mathbb{C}_n^{3\text{-}OP}$, explicit structural form through unitary diagonalization, and connections to Hermitian, MP, EP, PI, TM, SD, and related classes, as well as new linear and power-based equations linking $A$, $A^*$, and $A^\dagger$. The results provide a unified framework for recognizing and manipulating orthogonal tripotent matrices, with potential applications in operator theory and matrix equations and promising avenues for extending the theory to Hilbert spaces and dual-number settings.

Abstract

In this paper, we present different characterizations of tripotent orthogonal matrices (i.e., A^3 = A = A^* ) in terms of matrix equations, integer powers of AA^* and A^*A, average of A, A^*, and A^{\dagger}, rank of matrices, and trace of matrices. We study certain properties of this class of matrices.

Theorems & Definitions (24)

• Lemma 1.1
• Lemma 1.2
• Remark 1.3
• Remark 1.4
• Example 1.5
• Lemma 1.6
• Theorem 2.1
• Remark 2.2
• Theorem 2.3
• Theorem 2.4