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arxiv: 1907.09291 · v1 · pith:JSCLLVB2new · submitted 2019-07-22 · 🧮 math.NA · cs.NA

Reverse-order law for core inverse of tensors

Jajati Kesahri Sahoo , Ratikanta Behera This is my paper

Pith reviewed 2026-05-24 18:00 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords core inversetensorsEinstein productreverse-order lawmultilinear systemsgeneralized inversesPoisson problem
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The pith

Sufficient and necessary conditions are established for the reverse-order law of the core inverse of tensors under the Einstein product.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper works with the core inverse of tensors defined via the Einstein product, a notion introduced only recently. It derives conditions that are both necessary and sufficient for the reverse-order law to hold, so the core inverse of a product equals the product of the individual core inverses taken in reverse order. Additional results address mixed-type versions of this law. The core inverse is applied to obtain solutions for multilinear systems of tensor equations. An explicit demonstration solves the Poisson problem inside this multilinear framework.

Core claim

The notion of the core inverse of tensors with the Einstein product was introduced very recently. Sufficient and necessary conditions for the reverse-order law of this inverse are established. New results related to the mixed-type reverse-order law for core inverse are presented. Core inverse solutions of multilinear systems of tensors via the Einstein product are discussed, and the approach is demonstrated for solving the Poisson problem in the multilinear system framework.

What carries the argument

The core inverse of a tensor with respect to the Einstein product, together with the rank and range conditions that make the reverse-order law (A * B)^# = B^# * A^# hold.

If this is right

  • The reverse-order law holds exactly when the identified sufficient and necessary conditions on the tensors are met.
  • Mixed-type reverse-order laws hold under the additional relations derived in the paper.
  • Multilinear tensor systems possess core-inverse solutions whenever the core inverse exists.
  • The Poisson equation admits an explicit solution when recast as a multilinear system and solved via the core inverse.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The conditions may guide the design of algorithms that compute core inverses for large tensor arrays without forming the full product first.
  • Similar rank-based criteria could be tested for other generalized inverses of tensors under the same product.
  • The multilinear Poisson example suggests the method may apply to other linear PDEs discretized on tensor grids.
  • Numerical stability of the resulting solutions could be checked on low-order test tensors that meet the conditions.

Load-bearing premise

The core inverse of each tensor in the statements is assumed to exist.

What would settle it

A concrete pair of tensors A and B satisfying the stated rank and range conditions for which (A * B)^# does not equal B^# * A^#.

Figures

Figures reproduced from arXiv: 1907.09291 by Jajati Kesahri Sahoo, Ratikanta Behera.

Figure 1
Figure 1. Figure 1: Solution of the multilinear system for different values of [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗
read the original abstract

The notion of the core inverse of tensors with the Einstein product was introduced, very recently. This paper we establish some sufficient and necessary conditions for reverse-order law of this inverse. Further, we present new results related to the mixed-type reverse-order law for core inverse. In addition to these, we discuss core inverse solutions of multilinear systems of tensors via the Einstein product. The prowess of the inverse is demonstrated for solving the Poisson problem in the multilinear system framework.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper builds on the recently introduced core inverse for tensors under the Einstein product. It derives sufficient and necessary conditions for the reverse-order law ((AB)^# = B^# A^#) to hold, presents results on mixed-type variants of this law, and shows how the core inverse yields solutions to multilinear tensor systems, with a demonstration on the Poisson problem.

Significance. If the derivations hold, the work extends matrix core-inverse theory to the tensor setting with the Einstein product, supplying explicit conditions under which the reverse-order law applies and a concrete method for solving multilinear systems. This could be useful for structured tensor computations in applications such as discretized PDEs.

major comments (2)
  1. [Abstract, §1] Abstract and §1: All stated sufficient/necessary conditions for the reverse-order law and the mixed-type variants, as well as the core-inverse solution method for multilinear systems, are conditional on the core inverse existing for the tensors involved. No new existence criteria or independent verification of existence within the Einstein-product framework are supplied; the results therefore inherit the standing assumption from the cited prior reference without additional analysis.
  2. [Application section (Poisson problem)] The demonstration for the Poisson problem is presented as an application of the core-inverse solution method, but because existence is not re-derived or guaranteed inside the manuscript, the numerical example only illustrates the method where the imported existence criterion already holds.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Abstract, §1] Abstract and §1: All stated sufficient/necessary conditions for the reverse-order law and the mixed-type variants, as well as the core-inverse solution method for multilinear systems, are conditional on the core inverse existing for the tensors involved. No new existence criteria or independent verification of existence within the Einstein-product framework are supplied; the results therefore inherit the standing assumption from the cited prior reference without additional analysis.

    Authors: We agree that all results are conditional on the existence of the core inverses, as established in the cited prior reference introducing the core inverse for tensors under the Einstein product. The manuscript's focus is the derivation of necessary and sufficient conditions for the reverse-order law (and mixed-type variants) assuming existence, together with the application to multilinear systems; new existence criteria lie outside the stated scope. In the revised version we will explicitly restate the standing existence assumption in the abstract and §1 to make the conditional nature of the results fully transparent. revision: partial

  2. Referee: [Application section (Poisson problem)] The demonstration for the Poisson problem is presented as an application of the core-inverse solution method, but because existence is not re-derived or guaranteed inside the manuscript, the numerical example only illustrates the method where the imported existence criterion already holds.

    Authors: The Poisson-problem example is intended solely as an illustration of how the core-inverse solution method applies to a concrete multilinear system once the existence conditions from the prior reference are met. We will add a clarifying sentence in the application section noting that the numerical demonstration assumes the relevant core inverses exist according to the imported criterion. revision: yes

Circularity Check

0 steps flagged

No significant circularity; conditions derived independently under existence prerequisite

full rationale

The paper assumes the core inverse exists (as introduced in cited prior work) and derives sufficient/necessary conditions for reverse-order laws plus applications to multilinear systems. These are presented as mathematical results without any quoted reduction of the claimed laws to fitted parameters, self-definitions, or load-bearing self-citations that force the outcomes by construction. The existence assumption is a standard prerequisite rather than a circular step within this manuscript's derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is visible, so the ledger is necessarily incomplete. No free parameters are mentioned. The central claims rest on the prior definition of the core inverse and on the algebraic properties of the Einstein product, both treated as background.

axioms (2)
  • domain assumption The core inverse of a tensor (as defined in the cited recent work) exists for the tensors under consideration.
    The abstract presupposes this existence before stating conditions for the reverse-order law.
  • standard math The Einstein product obeys the usual associativity and distributivity rules needed for inverse manipulations.
    Implicit in any algebraic treatment of tensor inverses via this product.

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Reference graph

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