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arxiv: 1907.08408 · v1 · pith:K74CR3RCnew · submitted 2019-07-19 · 🧮 math.FA

Solutions of a class of nonlinear matrix equations

Samik Pakhira , Snehasish Bose , Sk Monowar Hossein This is my paper

Pith reviewed 2026-05-24 19:11 UTC · model grok-4.3

classification 🧮 math.FA
keywords nonlinear matrix equationsHermitian positive definite solutionsnecessary and sufficient conditionsmatrix iterationsmaximal solutionsminimal solutions
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The pith

Necessary and sufficient conditions determine existence of Hermitian positive definite solutions to the nonlinear matrix equation X^s + A^*X^{-t}A + B^*X^{-p}B = Q.

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

The paper establishes necessary and sufficient conditions under which the nonlinear matrix equation X^s + A^*X^{-t}A + B^*X^{-p}B = Q has Hermitian positive definite solutions when the exponents are at least one. These conditions require the coefficient matrices A and B to be nonsingular and the right-hand side Q to be Hermitian positive definite. The authors supply iterative procedures for computing the solutions and characterize the maximal and minimal ones among any that exist. A reader would care because the equation models relations involving matrix powers and inverses that appear in stability analysis and control problems.

Core claim

Several necessary and sufficient conditions are presented for the existence of Hermitian positive definite solutions of the nonlinear matrix equation X^s + A^*X^{-t}A + B^*X^{-p}B = Q, where s, t, p ≥ 1, A, B are nonsingular matrices and Q is a Hermitian positive definite matrix. Iterations are derived to compute the solutions, followed by examples, and the maximal and minimal Hermitian positive definite solutions are discussed.

What carries the argument

The nonlinear matrix equation X^s + A^*X^{-t}A + B^*X^{-p}B = Q together with the necessary and sufficient conditions on its parameters and matrices that guarantee Hermitian positive definite solutions.

If this is right

  • When the conditions hold, the derived iterations produce the solutions.
  • The largest and smallest Hermitian positive definite solutions can be distinguished and computed.
  • Existence is tied directly to relations among the given matrices and exponents.

Where Pith is reading between the lines

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

  • The iterations could be implemented directly as numerical solvers for related matrix problems.
  • The conditions might extend to cases with singular A or B if suitable limits or regularizations are introduced.

Load-bearing premise

The parameters satisfy s, t, p at least 1, the matrices A and B are nonsingular, and Q is Hermitian positive definite.

What would settle it

A concrete choice of s, t, p, A, B, and Q that satisfies the stated necessary and sufficient conditions yet has no Hermitian positive definite solution would falsify the claim.

Figures

Figures reproduced from arXiv: 1907.08408 by Samik Pakhira, Sk Monowar Hossein, Snehasish Bose.

Figure 1
Figure 1. Figure 1: Convergence history of (2.15) with error e = 3.124677099290812 × 10−16. Therefore equation (2.14) has a Hermitian positive definite solution in [α 1 3 I, Q1 3 ] as   1.257819473237711 −0.000309853059784 0.000829790450201 −0.000309853059784 1.253124679713870 0.001525655417243 0.000829790450201 0.001525655417243 1.256499440822798  . Now let t ≥ s and without loss of generality let t ≥ p, then max{s, t, p… view at source ↗
Figure 2
Figure 2. Figure 2: Convergence history of (2.20) References 1. W.N. Anderson, T.D. Morley, G.E. Trapp, Positive solutions to X = A − BX−1B ∗ , Linear Algebra Appl. 134 (1990), 53–62. 2. M. Asgari, B. Mousavi, Solving a class of nonlinear matrix equations via the coupled fixed point theorem, Appl. Math. Comput. 259 (2015), 364–373. 3. M. Berzig, X. Duan, B. Samet, Positive definite solution of the matrix equation X = Q−A ∗X −… view at source ↗
read the original abstract

In this article we present several necessary and sufficient conditions for the existence of Hermitian positive definite solutions of nonlinear matrix equations of the form $X^s + A^*X^{-t}A + B^*X^{-p}B = Q$, where $ s, t, p \geq 1$, $ A, B$ are nonsingular matrices and $Q$ is a Hermitian positive definite matrix. We derive some iterations to compute the solutions followed by some examples. In this context we also discuss about the maximal and the minimal Hermitian positive definite solution of this particular nonlinear matrix equation.

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 claims to establish several necessary and sufficient conditions for the existence of Hermitian positive definite solutions to the nonlinear matrix equation X^s + A^* X^{-t} A + B^* X^{-p} B = Q (with s, t, p ≥ 1, A and B nonsingular, Q Hermitian positive definite), derives iterations for computing such solutions, supplies examples, and discusses the maximal and minimal solutions.

Significance. If the stated conditions are correctly derived and free of gaps, the work would add concrete existence criteria and practical iteration schemes to the theory of nonlinear matrix equations, which appear in control, optimization, and operator theory. Explicit discussion of maximal/minimal solutions would be a useful feature for applications requiring bounds. No machine-checked proofs or reproducible code are mentioned.

major comments (2)
  1. [Abstract] Abstract: the manuscript asserts 'several necessary and sufficient conditions' but neither states the conditions explicitly nor indicates the proof strategy, the precise role of the hypotheses (nonsingularity of A, B; positive-definiteness of Q), or any error analysis. Without these, it is impossible to verify that the claimed necessity and sufficiency hold without post-hoc restrictions or circularity.
  2. [Abstract] Abstract: no numerical verification, convergence analysis of the proposed iterations, or comparison with existing methods for similar equations (e.g., Riccati or Lyapunov-type) is referenced, leaving the computational claims unassessable.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the comments. We respond point by point to the major comments below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the manuscript asserts 'several necessary and sufficient conditions' but neither states the conditions explicitly nor indicates the proof strategy, the precise role of the hypotheses (nonsingularity of A, B; positive-definiteness of Q), or any error analysis. Without these, it is impossible to verify that the claimed necessity and sufficiency hold without post-hoc restrictions or circularity.

    Authors: The abstract is a high-level summary. The necessary and sufficient conditions are stated explicitly in the body of the manuscript together with their proofs. The nonsingularity of A and B ensures the relevant inverses exist and remain Hermitian, while positive-definiteness of Q guarantees consistency with the left-hand side. The proofs rely on monotonicity properties of the involved matrix maps and fixed-point arguments on the cone of positive definite matrices; these steps are direct and do not involve circular reasoning. We are willing to lengthen the abstract to mention the main hypotheses and proof approach. revision: partial

  2. Referee: [Abstract] Abstract: no numerical verification, convergence analysis of the proposed iterations, or comparison with existing methods for similar equations (e.g., Riccati or Lyapunov-type) is referenced, leaving the computational claims unassessable.

    Authors: The manuscript derives the iterations and supplies concrete examples that illustrate their use on specific matrices. No separate convergence-rate analysis or systematic comparison with Riccati/Lyapunov solvers appears in the paper; the emphasis is on existence theory and the identification of maximal and minimal solutions. The examples provide direct numerical checks for the cases considered. revision: no

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper asserts necessary and sufficient conditions for Hermitian positive definite solutions to the matrix equation under explicit structural hypotheses (s, t, p ≥ 1, A and B nonsingular, Q Hermitian positive definite). These conditions are presented as mathematical statements derived from the equation form itself rather than from any fitted parameters, self-referential definitions, or load-bearing self-citations that reduce the result to its inputs by construction. Iterations for computing solutions and discussion of maximal/minimal solutions are derived separately as constructive methods. No quoted step equates a claimed existence condition to a tautology or renames a fitted quantity as a prediction. The derivation chain remains self-contained against external benchmarks of matrix analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities can be extracted beyond the parameter restrictions stated in the abstract.

axioms (1)
  • domain assumption s, t, p >= 1; A, B nonsingular; Q Hermitian positive definite
    These are the explicit hypotheses listed in the abstract under which the existence conditions are claimed to apply.

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Works this paper leans on

23 extracted references · 23 canonical work pages

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