Outer Bohemian Inverses: Structure and Parameterization

• Outer Bohemian inverses are generalized inverses of matrices with entries in {0, ±1} that satisfy XAX = X, preserving the Bohemian structure.
• They extend classical inverse theory into a discrete setting by imposing combinatorial constraints that allow exact counting and parameterization.
• Structured classifications (Classes I, II, and III) facilitate analysis via blockwise and rank-one representations, highlighting the role of support separation.

Searching arXiv for the specified paper and closely related work on outer inverses to ground the article in current literature. {"query":"(Chowdhry et al., 23 Aug 2025) Characterizations of structured Bohemian matrices and their Inner and Outer Bohemian Inverses", "max_results": 5} {"query":"(Chowdhry et al., 23 Aug 2025) Characterizations of structured Bohemian matrices and their Inner and Outer Bohemian Inverses", "max_results": 5, "source": "arxiv"} Attempting direct arXiv lookup by identifier and title. Outer Bohemian inverses are generalized inverses of Bohemian matrices—matrices whose entries lie in the population $\mathbb{P}=\{0,\pm1\}$—that themselves remain Bohemian. In the matrix-theoretic setting of "Characterizations of structured Bohemian matrices and their Inner and Outer Bohemian Inverses" (Chowdhry et al., 23 Aug 2025), an inner inverse, or $\{1\}$-inverse, of $A$ is a matrix $X$ satisfying $AXA=A$, while an outer inverse, or $\{2\}$-inverse, satisfies $XAX=X$. The corresponding Bohemian inverse sets are obtained by intersecting the usual generalized-inverse sets with $\mathbb{P}^{n\times m}$. The topic therefore lies at the intersection of generalized inverse theory, structured matrix theory, and discrete combinatorics: the central problem is not only whether inverses exist, but which inverses preserve the Bohemian population and how those sets can be exactly parameterized and counted (Chowdhry et al., 23 Aug 2025).

1. Formal setting and basic distinctions

Let $A\in \mathbb{P}^{m\times n}$ be a Bohemian matrix, with $\mathbb{P}=\{0,\pm1\}$. The standard generalized inverse language used in the literature defines $\{1\}$0 to be an inner inverse of $\{1\}$1 if

$\{1\}$2

and an outer inverse of $\{1\}$3 if

$\{1\}$4

The corresponding sets are denoted $\{1\}$5 and $\{1\}$6. Restricting to Bohemian inverses means

$\{1\}$7

Accordingly, an inner Bohemian inverse is a Bohemian matrix satisfying $\{1\}$8, and an outer Bohemian inverse is a Bohemian matrix satisfying $\{1\}$9 (Chowdhry et al., 23 Aug 2025).

A basic conceptual distinction emphasized in the literature is that outer inverses are more delicate because they form nonlinear, affine-algebraic families rather than the linear families typical of many inner-inverse descriptions. This distinction is decisive for Bohemian matrices, because the population constraint $A$0 is discrete and therefore interacts differently with linear and nonlinear parameterizations. In this setting, exact characterization and exact counting become part of the theory rather than a secondary computational issue (Chowdhry et al., 23 Aug 2025).

The paper organizes the subject around structured classes of Bohemian matrices. The first class is rank-one Bohemian matrices. More generally, rank-$A$1 Bohemian matrices are written as

$A$2

From this representation, three structured classes are defined. Class I requires pairwise disjoint coordinatewise supports on both sides; Class II requires disjointness on one side only; and Class III is a special subclass of Class II defined by the orthogonality condition

$A$3

The inclusions are

$A$4

2. Rank-one Bohemian matrices and complete outer-inverse descriptions

For rank-one Bohemian matrices, the theory becomes completely explicit. Any rank-one $A$5 can, up to permutation and diagonal Bohemian equivalence, be written in the form

$A$6

where $A$7 are permutation matrices, $A$8 are invertible diagonal Bohemian matrices, and $A$9 is either a full type I matrix $X$0 or a full type III matrix $X$1, or the corresponding form with zero rows appended. This reduction is important because, via unitary equivalence, the cardinalities of Bohemian inverse sets are preserved (Chowdhry et al., 23 Aug 2025).

If $X$2 has rank one, then every nonzero outer inverse has rank one and is of the form $X$3, with

$X$4

Equivalently,

$X$5

Thus, for rank-one matrices the entire nonzero outer-inverse set is parameterized by a single scalar constraint coupling the two factors $X$6 and $X$7 (Chowdhry et al., 23 Aug 2025).

For the full type-I matrix $X$8, this specializes to

$X$9

When $AXA=A$0, the cardinality of the nonzero Bohemian outer-inverse set is

$AXA=A$1

For the type-III matrix

$AXA=A$2

the complete outer-inverse description is

$AXA=A$3

and the Bohemian cardinality is

$AXA=A$4

A key consequence is that all rank-one Bohemian matrices with the same pattern of zero rows and zero columns have the same number of inner or outer Bohemian inverses, because the defining conditions are preserved under the permitted equivalences (Chowdhry et al., 23 Aug 2025).

3. Structured classes I, II, and III

The higher-rank theory is organized by the three structured classes introduced from sums of rank-one Bohemian outer products. Class I is the most restrictive: the coordinatewise supports of the $AXA=A$5 are pairwise disjoint and the supports of the $AXA=A$6 are pairwise disjoint. Up to permutation equivalence, Class I matrices are precisely generalized well-settled matrices, i.e. block diagonal matrices whose diagonal blocks are rank-one Bohemian matrices (Chowdhry et al., 23 Aug 2025).

Class II is larger. It requires disjointness on one side only, so up to permutation equivalence such matrices are row-blocked or column-blocked collections of rank-one Bohemian blocks. Class III is a special subclass of Class II defined by the orthogonality condition

$AXA=A$7

The inclusion chain

$AXA=A$8

expresses increasing structural flexibility while preserving enough organization to make explicit outer-inverse analysis possible (Chowdhry et al., 23 Aug 2025).

The significance of this classification is structural rather than merely taxonomic. For Class I and many Class II and Class III matrices, the inverse problem reduces to blockwise scalar constraints; for full-row-rank Class II matrices and rank-two Class III matrices, complete descriptions of outer inverses are obtained. This suggests that support separation and block orthogonality are the decisive mechanisms that turn a generally nonlinear generalized-inverse problem into an exact combinatorial description (Chowdhry et al., 23 Aug 2025).

4. Rank-one outer inverses in Classes I and II

For Class I and Class II, the analysis first concentrates on rank-one outer inverses. If $AXA=A$9 is block diagonal with rank-one blocks $\{2\}$0, then a rank-one outer inverse $\{2\}$1 is characterized by

$\{2\}$2

where $\{2\}$3 and $\{2\}$4 are partitioned compatibly with the block structure (Chowdhry et al., 23 Aug 2025).

In the pure Class I case, where each block is $\{2\}$5, the condition becomes

$\{2\}$6

For a Class II matrix of the form $\{2\}$7, where $\{2\}$8 is a pure well-settled block matrix, the outer-rank-one condition becomes

$\{2\}$9

with $XAX=X$0 (Chowdhry et al., 23 Aug 2025).

These formulas show that rank-one outer inverses in Class I and Class II are controlled by a single scalar constraint involving blockwise entry sums. In comparison with abstract generalized-inverse descriptions, the formulas expose a direct connection between block geometry and admissible Bohemian inverses. The dependence on the entry-sum functional $XAX=X$1 also shows that, in these classes, the outer-inverse condition can be expressed without leaving the discrete combinatorial language of the population $XAX=X$2 (Chowdhry et al., 23 Aug 2025).

5. Full-row-rank Class II matrices and rank-two Class III matrices

For full-row-rank Class II matrices of rank $XAX=X$3, the relevant outer inverses are actually $XAX=X$4-inverses, i.e. reflexive inverses. The complete characterization is

$XAX=X$5

This theorem says that for full-row-rank Class II matrices, the outer inverse condition forces the blockwise diagonal structure of the inverse and mutual annihilation between distinct blocks. In other words, the global $XAX=X$6-inverse problem reduces to independent $XAX=X$7-inverse problems on the rank-one blocks, together with orthogonality conditions across different blocks (Chowdhry et al., 23 Aug 2025).

For rank-two full-row-rank Class III matrices, the outer inverse set is split according to rank. If $XAX=X$8 is Class III, then

$XAX=X$9

The rank-one part $\mathbb{P}^{n\times m}$0 is described by the general rank-one outer-inverse formula above, while the rank-two part $\mathbb{P}^{n\times m}$1 is the $\mathbb{P}^{n\times m}$2-inverse set. For the basic well-settled case

$\mathbb{P}^{n\times m}$3

one has

$\mathbb{P}^{n\times m}$4

For other rank-two Class III patterns, the paper gives explicit sum-of-entries conditions such as

$\mathbb{P}^{n\times m}$5

together with analogous equations for the second block. The result is a full parameterization of outer inverses in the four structural types treated in the paper (Chowdhry et al., 23 Aug 2025).

6. Exact counting and broader outer-inverse context

A particularly explicit counting result is obtained for the outer Bohemian inverse set of a rank-two full-row-rank Class I matrix,

$\mathbb{P}^{n\times m}$6

In the Bohemian population $\mathbb{P}^{n\times m}$7,

$\mathbb{P}^{n\times m}$8

where

$\mathbb{P}^{n\times m}$9

This formula combines the contribution from rank-one outer Bohemian inverses with the contribution from rank-two reflexive inverses (Chowdhry et al., 23 Aug 2025).

Within generalized inverse theory, these results occupy a distinct position. Recent work on third-order tensors under the $A\in \mathbb{P}^{m\times n}$0-product characterizes outer inverses with prescribed range and kernel and derives FFT-based and $A\in \mathbb{P}^{m\times n}$1-QR-based computational formulas (Behera et al., 2023). Subsequent work under the more general $A\in \mathbb{P}^{m\times n}$2-product develops an $A\in \mathbb{P}^{m\times n}$3-QR construction and a factorized $A\in \mathbb{P}^{m\times n}$4-HPI19 iterative method with nineteenth-order convergence for outer inverses with prescribed range and kernel (Behera et al., 2024). In quaternion matrix theory, outer inverses and $A\in \mathbb{P}^{m\times n}$5-inverses with prescribed range and null space are characterized separately in the left and right senses, with explicit formulas and full-rank-decomposition representations (Bhadala et al., 24 Jun 2025).

This broader context suggests a common outer-inverse template—selection by algebraic identities together with structural constraints—but the Bohemian case is distinguished by the discrete population constraint $A\in \mathbb{P}^{m\times n}$6 and by exact combinatorial counting. In that sense, outer Bohemian inverses extend classical inverse theory into a discrete, combinatorial setting in which existence, parameterization, and cardinality are treated simultaneously (Chowdhry et al., 23 Aug 2025).