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Open research questions in Matrix Theory and Algorithms

145 gap statements mined from Matrix Theory and Algorithms papers in our 4.5M-paper local library, which holds 1,234 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.

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What the literature leaves open

  • Existing methods for solving the matrix equation AXB = C have high computational complexity and memory requirements. There is a need for an efficient method that can handle large-scale problems with structured or sparse coefficient matrices.

    Quaternionic conjugate-gradient method for solving the matrix equation $$AXB=C$$ over generalized quaternions · 2026 · DOI

  • Future research can focus on improving the efficiency and scalability of the proposed approach. The applicability of the approach to other types of data and applications can be explored.

    Fast Homomorphic Linear Algebra with BLAS · 2026 · DOI

  • The lack of efficient homomorphic linear algebra algorithms is a significant gap in existing solutions. The proposed approach addresses this gap by using CKKS and BLAS.

    Fast Homomorphic Linear Algebra with BLAS · 2026 · DOI

  • The geometric product generates higher-grade multivector terms, which could destroy the geometric interpretation of the iterated points. The traditional generation of Julia-type fractal sets relies on the iteration of complex functions, which is limited to the complex plane.

    Vector Invariance and Structural Closure of Julia-Type Iterations in Clifford Algebra · 2026

  • The existing results for the Drazin inverse of the sum P+Q have stronger assumptions than those presented in this paper. There is a need for explicit representations for Drazin inverses of complex partitioned matrices under weaker conditions. The paper identifies a gap in the existing literature and provides new results to fill this gap.

    A Note on Representations of the Drazin Inverse for Complex Partitioned Matrices · 2026 · DOI

  • The complete characterization of the Kronecker structure of a matrix pencil perturbed by another pencil of rank one involves very involved conditions. There is a need to better understand the meaning of those conditions.

    Bounds for the change of the Weyr characteristic of matrix pencils after 1-rank perturbations · 2026 · DOI

  • The Johnson-Newman theorem is limited to a single rank-one perturbation. The generalized theorem by Wang and Zhao has a complex condition for simultaneous similarity.

    On a generalization of the Johnson-Newman theorem to multiple rank-one perturbations · 2026 · DOI

  • The paper does not provide a comprehensive literature survey. The results are limited to the study of matrix tuples and do not generalize to more general settings.

    Spectral test of reducibility for matrix tuples · 2026 · DOI

  • The paper suggests that future research could focus on generalizing the results to more general settings. The paper notes that the study of common invariant subspaces is an active area of research and that new results are likely to emerge in the future.

    Spectral test of reducibility for matrix tuples · 2026 · DOI

  • The paper does not discuss the limitations of the P-Series in detail. The architecture may be limited by the assumption of box-shaped pixels. The paper does not provide a comprehensive evaluation of the P-Series in different scenarios.

    Phase-Encoded Analytic Series for Sublinear Spectral Array Analysis with Native Noise Annihilation · 2026 · DOI

  • The traditional discrete constraints of spatial frequency analysis are a limitation. Prior work in spatial encoding and spectral analysis has not fully addressed this limitation. The P-Series is designed to fill this gap and enable absolute interferometric annihilation of non-target spectral frequencies.

    Phase-Encoded Analytic Series for Sublinear Spectral Array Analysis with Native Noise Annihilation · 2026 · DOI

  • The authors suggest further study of the properties of spin Leonard pairs and their applications. The authors also suggest the study of the zero diagonal space of a Leonard pair and its relation to the spin property.

    Spin Leonard pairs and the zero diagonal space · 2026 · DOI

  • The authors identify a gap in the literature on Leonard pairs, specifically the lack of a necessary and sufficient condition for a Leonard pair to have spin. The authors also identify a gap in the classification of Leonard pairs that satisfy this condition.

    Spin Leonard pairs and the zero diagonal space · 2026 · DOI

  • They hold for the test problems presented in this paper and for many others that we have tried, but identifying this class of matrices remains an open problem. Identifying this region remains an open problem.

    When is the Resolvent Like a Rank One Matrix? · 2026 · DOI

  • The results demonstrate that classical Julia dynamics can be consistently extended beyond the complex plane into higher-dimensional geometric algebra.

    Vector Invariance and Structural Closure of Julia-Type Iterations in Clifford Algebra · 2026

  • The paper notes that the choice of preconditioner can significantly affect the accuracy of the solution.

    Stable algorithms for general linear systems by preconditioning the normal equations · 2026 · DOI

  • Further study on the applications of the dual C-S inverse and the special dual C-S inverse. Further study on the characterization of stronger i-EP matrices and stronger appreciable i-EP matrices.

    Dual C-S inverse and its applications · 2026 · DOI

  • The paper identifies a gap in the characterization of dual generalized inverses. The paper identifies a gap in the introduction of the dual C-S inverse and the special dual C-S inverse.

    Dual C-S inverse and its applications · 2026 · DOI

  • Further evaluation of the HKT method on more complex geometries and problems. Development of new methods for overcoming the instability of the linear systems. Application of the HKT method to other fields such as acoustics, electromagnetics, and elasticity.

    Efficient Krylov-regularization solvers for multiquadric RBF discretizations of the 3D Helmholtz equation · 2026 · DOI

  • The paper identifies a gap in the understanding of polynomial equations over matrices. The paper aims to fill this gap by studying the solution set S and calculating its dimension.

    Simple polynomial equations over (m x m)-matrices · 2026 · DOI

  • To further generalize the theorem to more general perturbations. To apply the generalized theorem to other fields such as graph theory and network analysis.

    On a generalization of the Johnson-Newman theorem to multiple rank-one perturbations · 2026 · DOI

  • The assumption that the duration of each iteration of the i-th machine is proportional to log_2 2k_i, - The optimization described by Brent must be applied to justify this assumption, - The range of k_i displayed in Figure 2…

    Choosing iteration maps for the parallel Pollard rho method · 2026 · DOI

  • Exploring the behavior of G for larger M, - Investigating the optimal iteration maps for more than one machine, - Considering the range of k_i for M ≥…

    Choosing iteration maps for the parallel Pollard rho method · 2026 · DOI

  • This invites further investigation into weaker sufficient conditions for stability, possibly through the use of alternative matrix inverse perturbation arguments.

    A Note on the Stability of the Sherman–Morrison–Woodbury Formula · 2026 · DOI

  • The paper identifies a gap in the study of the relations between powers, roots and Moore–Penrose inverses of square matrices and generalized Fibonacci numbers.

    On Powers, Roots and Moore–Penrose Inverses of Matrices Via Generalized Fibonacci Numbers · 2026 · DOI

Most-cited papers in Matrix Theory and Algorithms

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Open questions raised by papers listed on this page include: (1) There is a need for an efficient method that can handle large-scale problems with structured or sparse coefficient matrices (“Quaternionic conjugate-gradient method for solving the matrix equation…”, 2026); (2) Future research can focus on improving the efficiency and scalability of the proposed approach (“Fast Homomorphic Linear Algebra with BLAS”, 2026); (3) There is a need for explicit representations for Drazin inverses of complex partitioned matrices under weaker conditions (“A Note on Representations of the Drazin Inverse for Complex Partitioned Matrices”, 2026); (4) There is a need to better understand the meaning of those conditions (“Bounds for the change of the Weyr characteristic of matrix pencils after 1-rank…”, 2026); (5) The paper suggests that future research could focus on generalizing the results to more general settings (“Spectral test of reducibility for matrix tuples”, 2026). 145 gap statements have been mined from Matrix Theory and Algorithms papers in our 4.5M-paper local library, which holds 1,234 papers on the topic; the gaps come from whichever of those papers state one. They are mostly the research gaps the authors state and the papers' abstracts, plus future-work, limitations and challenges passages. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one, so you can read the original claim in context.

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