Non-Standard Pseudo-Inverses

Updated 9 November 2025

Non-Standard Pseudo-Inverses are generalized inverses that relax the classical four Penrose conditions to overcome invertibility obstacles in diverse algebraic structures.
Methodologies include altering algebraic relations to yield variants like {1,2}-inverses, pseudo core inverses, and one-sided inverses for tailored computational and structural advantages.
Applications span linear algebra, control theory, and non-commutative rings, enhancing efficiency in model reduction and spectral analysis.

A non-standard pseudo-inverse is any generalized inverse that deviates from the classical Moore–Penrose pseudoinverse, either by altering the defining algebraic relations, by targeting structural or computational desiderata (such as sparsity), or by adapting to the algebraic or geometric setting (e.g., *-rings, tropical/supertropical semirings, Lie algebra representations, or operator/polynomial algebras). Such objects have proliferated across fields including linear and multilinear algebra, non-commutative ring theory, algebraic semirings, control, high-dimensional statistics, and nonlinear functional analysis. The unifying theme is the relaxation, generalization, or context-specific adaptation of the four Penrose (Moore–Penrose) conditions, often to address invertibility obstructions, exploit problem structure, or enable efficient computation.

1. Algebraic Foundations and Typology

The canonical Moore–Penrose inverse $A^+$ of $A\in\mathbb{C}^{m\times n}$ is uniquely defined as the $X$ satisfying the Penrose equations: (P1) $AXA=A$ , (P2) $XAX=X$ , (P3) $(AX)^* = AX$ , (P4) $(XA)^* = XA$ . Non-standard pseudo-inverses arise by modifying one or more of these:

{1,2}-inverse: Satisfies only reflexivity and idempotence.
{1,2,3}-inverse: Adds symmetry of $AX$ (Hermitian).
{1,2,4}-inverse: Adds symmetry of $XA$ .
Minimal/weak Drazin inverse: Targets singular/radical elements in non-commutative rings, relaxing spectral constraints.
(Pseudo-)core inverses: Incorporate extra Hermitian or minimality conditions in *-rings.
Supertropical pseudo-inverses: Generalize to idempotent semirings with ghost-supplemented partial orders.
Sparse and block-structured pseudo-inverses: Enforce linear constraints plus sparsity-inducing term.

These generalizations are not merely technical curiosities—they address obstructions to invertibility (non-full rank, singularity, radical elements, indefinite context) or enable structure tailored for specific computational or algebraic tasks.

Let $R$ be a -ring (associative ring with involution). The **pseudo core inverse* $a^{\circledR}$ of $a\in R$ is defined via existence of $x$ and minimal $m$ such that:

$x a^{m+1} = a^m$ ,
$a x^2 = x$ ,
$(a x)^* = a x$ .

This class includes the core inverse ( $m=1$ ), and the matrix core-EP inverse (arbitrary index). Existence and uniqueness are characterized: such an $x$ is unique; $a$ is pseudo core invertible iff $a$ is Drazin invertible of index $m$ and $a^m$ admits a $\{1,3\}$ -inverse; $a^{\circledR} = a^D a^m (a^m)^{(1,3)}$ (Gao et al., 2016).

Related are the right and left pseudo core inverses (resp. $a^{\scriptsize\textcircled{\tiny D}_r}$ and $a^{\scriptsize\textcircled{\tiny D}_l}$ ), which enforce core-like identities on only one side (Wang et al., 2018): $\text{Right: } a x a^k = a^k,\ x = a x^2,\ (a x)^* = a x.$ Module-theoretic characterizations relate pseudo core invertibility to powers of $a$ and their right/left ideals. These one-sided classes interpolate between group-invertibility, Moore–Penrose invertibility, and EP membership.

Pivotal algebraic laws extend: reverse-order, absorption, and additivity. For example,

$(ab)^{\circledR} = b^{\circledR} a^{\circledR} \quad \text{when } a,b \text{ commute and } a b^* = b^* a.$

$(a+b)^{\circledR} = a^{\circledR} + b^{\circledR} \quad \text{when } ab=ba=0 \text{ and } a^*b=0.$

These results provide explicit computational pathways and underpin a refined taxonomy of ring-theoretic inverses.

Given $a, f\in R$ with pseudo core inverses $a^{\scriptsize\textcircled{\tiny D}}$ , $f^{\scriptsize\textcircled{\tiny D}}$ , the absorption law (Zhou et al., 6 Aug 2025) states: $a^{\scriptsize\textcircled{\tiny D}}(a+f)f^{\scriptsize\textcircled{\tiny D}} = a^{\scriptsize\textcircled{\tiny D}} + f^{\scriptsize\textcircled{\tiny D}}$ if and only if $1 + a^{\scriptsize\textcircled{\tiny D}} b$ (where $b = f - a$ ) is invertible and $f^{\scriptsize\textcircled{\tiny D}} = (1 + a^{\scriptsize\textcircled{\tiny D}} b)^{-1} a^{\scriptsize\textcircled{\tiny D}}$ (the additive property).

Parallel absorption/additive laws hold for the minimal weak Drazin inverse, group inverse, (one-sided) core and core-EP inverses, and dual pseudo core inverses, with the invertibility of $1+a^* b$ (with starred variant for each context) the central algebraic condition.

In the context of matrices, the classical core–nilpotent and Schur decompositions yield tractable block expressions for $A^{\textcircled{D}}$ , and the absorption/additive laws translate to factorizations reducing to core/nilpotent and invertible/nilpotent parts.

4. Connections to Other Non-Standard Pseudoinverses

Non-standard pseudo-inverses abound in the literature, distinguished by which subset of Penrose equations or algebraic surrogates are enforced. Table 1 summarizes core classes.

Pseudo-inverse Type	Defining Relations	Algebraic/Computational Role
Moore–Penrose	(P1)-(P4)	Least-squares & minimal-norm solutions
Minimal weak Drazin	Pseudo core + index	Generalizations in rings, weak Drazin classes
Core/EP/One-sided Core	Pseudo core $m=1$ or module conditions	Hilbert-space projections, *-rings
Sparse (block/row versions)	Minimize norm, subset (P1)-(P4)	Efficient computation, model/interp.
Supertropical adjugate/closure	$\operatorname{adj}(A)/\det(A)$	Idempotent semirings, combinatorial closure
Drazin & pseudo n-strong Drazin	$xax=x$ , commutation, radical membership	Radical/periodic ring structure

Key distinctions include the presence of involution, the Hermitian symmetry constraint, commutation with the original element, and closure under ring-theoretic operations.

5. Computational Approaches and Applications

Algorithms for pseudo core inverses and their variants rely on canonical decompositions and explicit formulas. For $A\in\mathbb{C}^{n\times n}$ , two approaches are prevalent (Gao et al., 2016):

Hartwig–Spindelböck block diagonalization: Reduction to invertible and nilpotent blocks, then computation via Drazin and $\{1,3\}$ -inverses.
Regular–Nilpotent Decomposition: Similarity to block diagonal form, followed by direct inversion on the regular component and zero extension.

In the matrix setting, these permit direct evaluation of absorption/additive formulas when blocks commute and invertibility/nilpotence are preserved under perturbations (addition).

Applications include:

Efficient representation and computation in large or structured systems,
Model reduction and interpretability in data analysis,
Construction of associative or idempotent operations on functions with plateaus or discontinuities (e.g., via weak pseudo-inverses in monotone function spaces (Chen et al., 27 Oct 2024)),
Analysis and control in algebraic or Lie-algebraic settings, such as robotics or kinematics (see hyperbolic pseudoinverses (Donelan et al., 2017)).

6. Generalizations in Non-commutative, Radical, and Supertropical Settings

The theory extends robustly to non-classical rings (e.g., with radical elements), semirings, or contexts with partial or ghost-supplemented orders. For instance, the pseudo n-strong Drazin inverse (Cui et al., 2023) interpolates between Drazin and pseudo-Drazin invertibility: $a$ is pseudo n-strong Drazin invertible iff it is p-Drazin invertible and $a-a^{n+1}\in\sqrt{J(R)}$ , with equivalent idempotent-based characterizations. Extended Cline–Jacobson formulas and direct computation in block or triangular matrix rings illustrate the adaptability of the framework.

In supertropical algebra (Niv, 2013), the pseudo-inverse $A^\nabla = \operatorname{adj}(A)/\det(A)$ satisfies tropical analogues of classical identities (e.g., determinant formulae), but also realizes new phenomena: it is an involution (up to ghost equivalence), relates to matrix closure/stabilization, and induces eigenvalue reversal in characteristic polynomials.

7. Open Problems and Theoretical Directions

Current challenges include:

Full characterization of the conditions under which reverse-order laws for generalized inverses hold, especially in rings or semirings (Kędzierski, 3 Apr 2024).
Criteria and efficient algorithms for the additive property when orthogonality constraints are relaxed.
Continuity, stability, and spectral theory of pseudo core inverses under perturbations or in infinite-dimensional settings (e.g., $C^*$ -algebras).
The development of weighted variants and their implications for regularization and statistical consistency.
Equivariant constructions and invariance under group actions, particularly in geometric, control, or statistical settings.

A plausible implication is that as applications and structure-specific requirements increase (e.g., data sparsity, block-structure, operator semigroups, radical/thin invertibility conditions), the taxonomy and algorithmics of non-standard pseudo-inverses will continue to expand and diversify, with algebraic, combinatorial, and analytic tools being further integrated across domains.