Weighted Core-EP Inverse Overview

• Weighted Core-EP Inverse is a generalization of the core-EP inverse that extends index-one constructions to arbitrary-index settings using weighted metrics.
• It employs different formulations for square matrices, rectangular matrices, and ring elements by incorporating weighted Hermitian symmetry and projector constraints.
• This framework isolates the stable, invertible part of a matrix relative to a prescribed weight, offering robust decomposition and efficient computation of generalized inverses.

Searching arXiv for recent and foundational papers on weighted core-EP inverse and closely related weighted core inverse / weighted-EP notions. The weighted core-EP inverse is a weighted generalization of the core-EP inverse designed to extend core-inverse-type constructions from index-one matrices to arbitrary-index settings, while incorporating an external weight through weighted Hermitian or projector conditions. In the literature, the term appears in several closely related but not identical frameworks: for square matrices with positive or invertible Hermitian weights, for rectangular matrices with a weighting matrix $W$, and, indirectly, in rings with involution through the theory of the weighted core inverse (or $e$-core inverse) and weighted-EP elements (Behera et al., 2020, Gao et al., 2018, Li, 2021). Across these settings, the unifying theme is that the inverse is characterized by a Drazin- or group-inverse component together with weighted orthogonality, weighted range restrictions, or weighted projector identities. This suggests a common structural principle: the weighted core-EP inverse isolates the stable, invertible part of an element or matrix relative to a weighted geometry, while suppressing the nilpotent or defect component (Wang, 2016, Ferreyra et al., 2024).

1. Terminological and conceptual setting

The expression weighted core-EP inverse is used in at least three technical senses in the cited literature.

First, for square complex matrices with arbitrary index $k=\operatorname{ind}(A)$, the paper "Further results on weighted core-EP inverse of matrices" defines the $E$-weighted core-EP inverse $A^{,E}$ by the equations

$XA^{k+1}=A^k,\qquad AX^2=X,\qquad (EAX)^*=EAX,$

and defines the dual notion $A^{F,}$ analogously (Behera et al., 2020). This is the most direct weighted analogue of the ordinary core-EP inverse.

Second, for rectangular matrices $A\in\mathbb C^{m\times n}$ with a nonzero weighting matrix $W\in\mathbb C^{n\times m}$, the $W$-weighted core-EP inverse $e$0 is characterized by

$e$1

where $e$2 (Gao et al., 2018). Equivalent forms involve Moore–Penrose inverses and weighted outer-inverse formulations.

Third, in rings with involution, the closest analogue is the weighted core inverse or $e$3-core inverse, defined for $e$4 and invertible Hermitian $e$5 by the existence of $e$6 satisfying

$e$7

The same paper then characterizes weighted-EP with respect to $e$8 by simultaneous compatibility of two such weighted core structures (Li, 2021). Although this is not itself a weighted core-EP inverse of arbitrary index, it provides the ring-theoretic precursor and the main weighted-EP structural link.

A plausible implication is that the phrase “weighted core-EP inverse” should be understood as a family resemblance rather than a single universal definition. The matrix literature splits between square weighted-by-Hermitian-metric formulations and rectangular weighted-by-operator formulations, while the ring literature supplies the projection/idempotent interpretation that underlies weighted-EP theory (Behera et al., 2020, Gao et al., 2018, Li, 2021).

2. Definitions and principal formulations

In the square-matrix framework, the $e$9-weighted core-EP inverse of $k=\operatorname{ind}(A)$0 with $k=\operatorname{ind}(A)$1 is the unique matrix $k=\operatorname{ind}(A)$2 such that

$k=\operatorname{ind}(A)$3

When $k=\operatorname{ind}(A)$4, these equations reduce to the weighted core inverse conditions, so the construction genuinely extends the index-one theory (Behera et al., 2020).

A key equivalent characterization states that $k=\operatorname{ind}(A)$5 if and only if

$k=\operatorname{ind}(A)$6

Thus the inverse is an outer inverse with prescribed range and weighted adjoint-range constraints (Behera et al., 2020). This range-based viewpoint is central: it shows that weighting modifies not the Drazin-type stabilization condition $k=\operatorname{ind}(A)$7, but the symmetry and geometric side of the inverse.

For rectangular matrices, the $k=\operatorname{ind}(A)$8-weighted core-EP inverse is defined by a projector equation rather than a weighted Hermitian equation: $k=\operatorname{ind}(A)$9 equivalently

$E$0

with $E$1 (Gao et al., 2018, Kara et al., 8 Jan 2025). The same literature gives the formula

$E$2

and notes that when $E$3 and $E$4, this reduces to the usual core-EP inverse (Ferreyra et al., 2024).

In rings with involution, the $E$5-core inverse is defined through principal one-sided ideal conditions and weighted adjoint compatibility, and the main theorem states that $E$6 is $E$7-core invertible if and only if there exists a unique idempotent $E$8 such that

$E$9

for arbitrary $A^{,E}$0 (Li, 2021). This gives the weighted ring-theoretic counterpart of projector-based core-EP structure.

The unweighted background is the ordinary core-EP inverse, defined for $A^{,E}$1 with $A^{,E}$2 by the unique $A^{,E}$3 satisfying

$A^{,E}$4

or equivalently by block decomposition and Drazin/Moore–Penrose formulas (Gao et al., 2021, Wang, 2016). The weighted theories preserve this stabilization mechanism while altering the selfadjointness or projection conditions.

3. Algebraic characterizations and representation formulas

One of the central achievements of the square-matrix weighted theory is its conversion into representation formulas through weighted $A^{,E}$5-inverses, Drazin inverses, and weighted Moore–Penrose inverses. If $A^{,E}$6 and $A^{,E}$7, then

$A^{,E}$8

and if $A^{,E}$9 exists, then

$XA^{k+1}=A^k,\qquad AX^2=X,\qquad (EAX)^*=EAX,$0

The same paper also gives the identities

$XA^{k+1}=A^k,\qquad AX^2=X,\qquad (EAX)^*=EAX,$1

for $XA^{k+1}=A^k,\qquad AX^2=X,\qquad (EAX)^*=EAX,$2 when the corresponding inverses exist (Behera et al., 2020).

The rectangular $XA^{k+1}=A^k,\qquad AX^2=X,\qquad (EAX)^*=EAX,$3-weighted theory is computationally richer. A foundational formula is

$XA^{k+1}=A^k,\qquad AX^2=X,\qquad (EAX)^*=EAX,$4

together with equivalent one-pseudoinverse representations

$XA^{k+1}=A^k,\qquad AX^2=X,\qquad (EAX)^*=EAX,$5

for $XA^{k+1}=A^k,\qquad AX^2=X,\qquad (EAX)^*=EAX,$6 (Gao et al., 2018). The same work identifies $XA^{k+1}=A^k,\qquad AX^2=X,\qquad (EAX)^*=EAX,$7 as a $XA^{k+1}=A^k,\qquad AX^2=X,\qquad (EAX)^*=EAX,$8-inverse of $XA^{k+1}=A^k,\qquad AX^2=X,\qquad (EAX)^*=EAX,$9 with prescribed range and nullspace, and gives full-rank, SVD, GAS, and QR-based representations (Gao et al., 2018).

The quaternion/complex determinantal approach extends the weighted core-EP inverse further by giving entrywise formulas for the right and left $A^{F,}$0-weighted core-EP inverses. In that setting,

$A^{F,}$1

and direct determinantal representations are derived using row and column determinants over quaternions, with ordinary determinant specializations in the complex case (Kyrchei, 2020).

In rings with involution, explicit formulas for the weighted core inverse arise from the unit/idempotent data. If $A^{F,}$2, then

$A^{F,}$3

Parallel formulas hold for the $A^{F,}$4 and non-idempotent $A^{F,}$5 versions (Li, 2021). This suggests that the ring theory isolates the same mechanism found in matrix weighted core-EP formulas: a power $A^{F,}$6 is corrected by a defect idempotent or projector to become invertible, and the inverse is then extracted on the stable component.

4. Decomposition principles and structural interpretation

The ordinary core-EP theory is governed by the core-EP decomposition, which writes a matrix $A^{F,}$7 of index $A^{F,}$8 as

$A^{F,}$9

where $A\in\mathbb C^{m\times n}$0 is core-invertible, $A\in\mathbb C^{m\times n}$1, and $A\in\mathbb C^{m\times n}$2 (Wang, 2016). The core-EP inverse is then simply

$A\in\mathbb C^{m\times n}$3

Equivalently, in unitary block form,

$A\in\mathbb C^{m\times n}$4

or, in another formulation,

$A\in\mathbb C^{m\times n}$5

depending on the specific decomposition convention used in the cited work (Gao et al., 2021, Wang, 2016). The common point is that only the nonsingular block survives in the inverse.

The weighted rectangular literature explicitly imports this idea through a weighted core-EP decomposition of the pair $A\in\mathbb C^{m\times n}$6. There exist unitary matrices $A\in\mathbb C^{m\times n}$7 and blocks $A\in\mathbb C^{m\times n}$8 such that

$A\in\mathbb C^{m\times n}$9

with $W\in\mathbb C^{n\times m}$0 and $W\in\mathbb C^{n\times m}$1 nilpotent of indices $W\in\mathbb C^{n\times m}$2 and $W\in\mathbb C^{n\times m}$3, respectively (Ferreyra et al., 2024). In the core-EP case $W\in\mathbb C^{n\times m}$4, the generalized $W\in\mathbb C^{n\times m}$5-BT family collapses to

$W\in\mathbb C^{n\times m}$6

This is the weighted analogue of “invert the stable block, annihilate the nilpotent block” (Ferreyra et al., 2024).

The same decomposition principle underlies the newer $W\in\mathbb C^{n\times m}$7-weighted $W\in\mathbb C^{n\times m}$8-BT inverse, where

$W\in\mathbb C^{n\times m}$9

and $W$0 yields $W$1 (Ferreyra et al., 2024). This parameterized interpolation clarifies the position of the weighted core-EP inverse within a wider projector-based family.

A plausible implication is that decomposition, rather than any single formula, is the most robust conceptual core of weighted core-EP theory. The weight determines the geometry of the stable block and the projector, but the inverse itself is still the inverse of the core part and zero on the nilpotent remainder.

5. Connections with weighted EP, weighted core inverse, and related generalized inverses

In ring theory, the weighted-EP condition provides a direct conceptual bridge. For invertible Hermitian weights $W$2, an element $W$3 is weighted-EP with respect to $W$4 if

$W$5

equivalently,

$W$6

The paper then proves that $W$7 is weighted-EP with respect to $W$8 if and only if there exists a unique idempotent $W$9 such that

$e$00

for arbitrary $e$01 (Li, 2021). This is a simultaneous two-weight analogue of the one-weight $e$02-core characterization.

In matrix theory, the square weighted core-EP inverse sits among several adjacent notions: weighted core inverse, weighted dual core inverse, generalized weighted Moore–Penrose inverse, Drazin inverse, and weighted-EP matrices. The range characterization

$e$03

shows that the weighted core-EP inverse is a weighted outer inverse; the formula

$e$04

links it to the Drazin inverse; and the use of $e$05 links it to the generalized weighted Moore–Penrose inverse (Behera et al., 2020).

The rectangular $e$06-weighted core-EP inverse is likewise connected to the weighted Drazin inverse by projector identities. One paper states

$e$07

showing that the weighted Drazin inverse and weighted core-EP inverse differ by the choice of orthogonal versus oblique projection onto the stabilized range (Gao et al., 2018).

The newer generalized-inverse-with-respect-to-$e$08 framework introduces

$e$09

for square matrices $e$10, and shows that the unweighted BT inverse and core-EP inverse arise as $e$11 and $e$12, while the $e$13-weighted BT and $e$14-weighted core-EP inverses appear as special weighted instances (Kara et al., 8 Jan 2025). This suggests that weighted core-EP inverses can be interpreted as members of a more general family of projector-compressed inverses.

6. Variants, extensions, and applications

The literature also develops parameterized variants that reduce to the weighted core-EP inverse in limiting cases. The $e$15-weighted $e$16-weak core inverse is defined by

$e$17

where $e$18 is the $e$19-weighted $e$20-weak group inverse, and the paper states that if $e$21, then

$e$22

identified there as the $e$23-weighted core-EP inverse (Ferreyra et al., 2024). The same paper provides multiple characterizations, including

$e$24

with unique solution $e$25, and derives a canonical form from a simultaneous unitary block upper triangularization (Ferreyra et al., 2024).

The $e$26-weighted $e$27-BT inverse

$e$28

interpolates between $e$29 at $e$30, the $e$31-weighted BT inverse at $e$32, and the $e$33-weighted core-EP inverse for $e$34 (Ferreyra et al., 2024). This shows that the weighted core-EP inverse occupies the stabilized end of a projector-indexed hierarchy.

In applications, the unweighted core-EP inverse has been used to solve fuzzy linear systems by converting the problem to a crisp associated system $e$35, with solvability criterion

$e$36

and generalized-solution constructions for inconsistent systems (Gao et al., 2021). The paper does not define a weighted core-EP inverse, but explicitly states that its block decomposition and solution formulas “naturally suggest weighted generalizations” (Gao et al., 2021). This suggests that weighted core-EP inverses may serve as weighted regularization-like operators in structured or inconsistent linear systems, although that application is not established directly in the cited weighted papers.

The Banach-algebra paper on generalized weighted EP elements does not define a matrix-style weighted core-EP inverse, but develops a decomposition

$e$37

and a projection criterion

$e$38

which are structurally close to core-EP decomposition ideas (Chen et al., 14 Jul 2025). This suggests a broader operator-algebraic direction for weighted core-EP-type theories beyond matrices.

7. Historical position and open structural themes

Historically, the weighted core-EP inverse emerged from two converging lines of work. One line extended the core inverse to arbitrary index through the core-EP inverse and its decomposition theory (Wang, 2016). The second line introduced weighting into core and Moore–Penrose frameworks, first for weighted core inverse in rings and square matrices, then for rectangular weighted core-EP constructions based on a matrix $e$39 (Li, 2021, Behera et al., 2020, Gao et al., 2018).

The literature shows several recurring themes.

Theme	Representative formulation	Source
Weighted Hermitian symmetry	$e$40	(Behera et al., 2020)
Weighted projector characterization	$e$41	(Gao et al., 2018)
Defect idempotent criterion	$e$42	(Li, 2021)

A recurring misconception is to treat all of these as the same object. The papers do not support that simplification. The square $e$43-weighted core-EP inverse, the rectangular $e$44-weighted core-EP inverse, and the ring-theoretic $e$45-core inverse/weighted-EP theory are parallel but non-identical constructions. They share stabilization by powers, weighted selfadjointness or projector conditions, and decomposition into stable versus nilpotent parts, but their definitions live in different categories and use different geometric data (Behera et al., 2020, Gao et al., 2018, Li, 2021).

Another important point is that weighted theories are not merely cosmetic modifications of the unweighted theory. The $e$46-weighted $e$47-BT paper explicitly states that identities valid for the $e$48-weighted Drazin inverse and $e$49-weighted core-EP inverse do not generally hold for intermediate $e$50-BT inverses when $e$51, showing that projector-based parameterization introduces genuinely new behavior (Ferreyra et al., 2024). Likewise, the ring paper notes that a tempting criterion involving $e$52 is not sufficient in general without a Dedekind-finite hypothesis (Li, 2021).

Taken together, the papers suggest that the weighted core-EP inverse should be understood as a weighted stable-part inverse: it is determined by powers of the operator or element, by a weighted geometry encoded in Hermitian metrics or weighting matrices, and by projections or idempotents that isolate the range on which inversion is meaningful. Its modern theory spans explicit formulas, canonical decompositions, range-nullspace descriptions, determinantal representations, and links to weighted EP and Drazin frameworks (Behera et al., 2020, Gao et al., 2018, Ferreyra et al., 2024, Li, 2021).