Characterizations of structured Bohemian matrices and their Inner and Outer Bohemian Inverses

TL;DR

This paper systematically characterizes various classes of Bohemian matrices with entries in {0, ±1}, focusing on their inner and outer Bohemian inverses, providing complete descriptions, cardinalities, and explicit formulas for specific classes and ranks.

Contribution

It introduces new characterizations and complete descriptions of inner and outer Bohemian inverses for multiple classes of Bohemian matrices, including explicit formulas and cardinalities.

Findings

Complete description of outer Bohemian inverse sets for rank-one matrices.

Characterizations of inner inverses for Class III and full-row rank Class II matrices.

Explicit formula for the cardinality of outer inverses for rank-two full-row rank Class I matrices.

Abstract

In this paper, we systematically define and characterize various classes of Bohemian matrices with respect to the population $\mathbb{P}=\{0, \pm 1\}$, focusing on their inner and outer Bohemian inverses. The classes under consideration include rank-one Bohemian matrices, as well as higher-rank Bohemian matrix classes, specifically Classes I, II, and III. For rank-one Bohemian matrices, a complete description of the outer Bohemian inverse sets is provided along with their cardinalities. Additionally, new insights into inner Bohemian inverses and their cardinalities are given. Characterizations of the complete set of inner inverses for the Class III matrices and full-row rank Class II matrices are obtained. Furthermore, we study the sets of rank-one outer inverses for Classes I and II, and examine the sets of rank $r$ outer inverses for full-row rank Class II matrices of rank $r$. Moreover, the set of outer inverses is completely characterized for rank-two full-row rank Class III matrices. In particular, we provide an explicit formula for the cardinality of the set of outer Bohemian inverses for rank-two full-row rank Class I matrices.