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arxiv: 2604.21818 · v1 · submitted 2026-04-23 · 🧮 math.NA · cs.NA

Formulae for the Drazin inverse of Modified Tensors via the Einstein Product

Yue Zhao , Daochang Zhang , Dijana Mosic This is my paper

Pith reviewed 2026-05-09 20:47 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Drazin inverseEinstein productmodified tensorgeneralized Schur complementSherman-Morrison-Woodbury formulahigher-order tensortensor equation
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The pith

The Drazin inverse of the modified tensor A minus C star_N D^D star_N B equals an expression built from A^D and the generalized Schur complement of D.

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

The paper derives closed-form expressions for the Drazin inverse of a block-modified higher-order tensor using the Einstein product. These expressions are written directly in terms of the Drazin inverse of one factor tensor and the generalized Schur complement formed from the remaining factors. The result unifies earlier matrix and tensor formulas and recovers the classical Sherman-Morrison-Woodbury formula when all tensors are of order two. A reader would care because the formula supplies an explicit, computable route to the Drazin inverse without requiring the solution of large auxiliary systems. The derivations rest on the algebraic rules of the Einstein product and the defining properties of the Drazin inverse.

Core claim

The paper establishes that the Drazin inverse of the modified tensor A - C *_N D^D *_N B is given by A^D plus correction terms involving the Drazin inverse of the generalized Schur complement D - B *_N A^D *_N C, together with the auxiliary tensors C and B. The same pattern yields a companion formula when the roles of the blocks are interchanged. These identities hold for tensors of arbitrary order provided the Drazin inverses exist.

What carries the argument

The Einstein product *_N acting on the block-modified expression together with the Drazin inverse operator applied to both the leading tensor and the generalized Schur complement.

If this is right

  • The new expressions directly generalize all previously known Drazin-inverse formulae for rank-one or low-rank modifications of matrices and second-order tensors.
  • When the tensors reduce to ordinary matrices the identities collapse to the classical Sherman-Morrison-Woodbury formula for the Drazin inverse.
  • The formulae supply an immediate way to update the Drazin inverse after a low-rank tensor modification without recomputing from scratch.
  • They extend the range of solvable tensor equations that can be treated by explicit algebraic manipulation rather than iterative methods.

Where Pith is reading between the lines

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

  • The same block-algebraic pattern may extend to other generalized inverses such as the Moore-Penrose or group inverse under suitable regularity conditions.
  • Efficient numerical implementations could be built by pre-computing the Drazin inverse of the leading tensor and then updating only the Schur complement when the modification tensors change.
  • The formulae might simplify the analysis of tensor networks or multilinear systems that arise in data science and physics when low-rank updates appear.

Load-bearing premise

The Drazin inverses of the participating tensors must exist and the Einstein product must satisfy the associative and distributive laws needed to carry out the block manipulations without extra index restrictions.

What would settle it

A concrete numerical example of tensors A, B, C, D of order three or higher in which the candidate formula produces a tensor that fails to satisfy at least one of the three defining equations of the Drazin inverse.

read the original abstract

This paper establishes exact expressions for the Drazin inverse of the modified tensor $\mathcal A-\mathcal C*_N\mathcal D^D*_N\mathcal B$ via the Einstein product, formulated using the Drazin inverse of $\mathcal A$ and the generalized Schur complement $\mathcal D-\mathcal B*_N\mathcal A^{D}*_N\mathcal C$, providing a comprehensive generalization and unification of existing results in the literature for the case when the tensors are of order two. Furthermore, the findings reduce to the classical Sherman-Morrison-Woodbury formula in the special case of second-order tensors. Finally, we give an example to illustrate our new explicit expression.

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

0 major / 0 minor

Summary. The paper derives explicit formulae for the Drazin inverse of the modified tensor A - C *_N D^D *_N B via the Einstein product. The expressions are given in terms of the Drazin inverse of A and the generalized Schur complement D - B *_N A^D *_N C. The work claims these formulae generalize and unify known results for order-two tensors, reduce to the classical Sherman-Morrison-Woodbury formula in the second-order case, and are supported by an illustrative example.

Significance. If the algebraic derivations hold, the results provide a direct and useful generalization of matrix Drazin-inverse identities to higher-order tensors under the Einstein product. This unifies existing literature on order-two cases and recovers classical formulae as special cases, which strengthens the contribution for researchers working on tensor generalizations of linear-algebraic inverses.

Simulated Author's Rebuttal

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We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The summary accurately reflects the paper's contributions: explicit formulae for the Drazin inverse of the modified tensor via the Einstein product, expressed in terms of the Drazin inverse of A and the generalized Schur complement, with unification of order-two results and reduction to the classical Sherman-Morrison-Woodbury formula.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents algebraic formulae for the Drazin inverse of the modified tensor A−C∗N D^D ∗N B expressed via the Drazin inverse of A and the generalized Schur complement, derived directly from the definitions of the Drazin inverse and the Einstein product. These are exact identities under the stated existence assumptions and reduce to known matrix results (including Sherman-Morrison-Woodbury) as special cases. No self-definitional loops, fitted inputs renamed as predictions, load-bearing self-citations, uniqueness theorems imported from the authors' prior work, or smuggled ansatzes appear; the derivation chain is self-contained within standard tensor algebra and does not reduce any claimed result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard definitions and algebraic closure properties of the Drazin inverse and Einstein product for tensors, which are drawn from prior literature without new free parameters or postulated entities.

axioms (1)
  • standard math The Einstein product is associative and compatible with Drazin inverse operations for tensors of compatible orders.
    Invoked when formulating the inverse of the modified tensor expression.

pith-pipeline@v0.9.0 · 5411 in / 1197 out tokens · 56647 ms · 2026-05-09T20:47:02.411150+00:00 · methodology

discussion (0)

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