Science AI Journal
Computer Science · Research topic

Open research questions in Matrix Theory and Algorithms

60 unresolved questions extracted from the limitations and future-work sections of 557 Matrix Theory and Algorithms papers in our library. Each links back to the study that raised it.

What the literature leaves open

  • Existing QCUR methods may suffer from heavy computational costs for large-scale quaternion problems. The need for a novel QCUR decomposition method that can efficiently handle large-scale quaternion matrices.

    A Randomized CUR Decomposition Based on Real Projectors for Large-scale Quaternion Matrices · 2026 · DOI

  • There is a gap in the understanding of the geometric regularities of sequences of matrix spectra. The paper identifies the need to understand the systematic cancellations coming from the symmetries of the matrices.

    Coherent sequences of matrix spectra, draft 7aug26 v3 · 2026 · DOI

  • The common invariant cone problem is known to be algorithmically undecidable for general families of matrices. There is a need for an efficient algorithm to solve the problem in many cases.

    Constructive Solution of the Common Invariant Cone Problem · 2026 · DOI

  • The paper is based on numerical experiments and does not provide a rigorous mathematical analysis of the results. The sample size is limited to the first 500 spectra associated with several intrinsically interesting sequences. The paper does not provide a systematic study of the behavior of matrix spectra associated with rational sequences.

    Coherent sequences of matrix spectra, draft 11aug26 v3 · 2026 · DOI

  • Future research could involve a rigorous mathematical analysis of the results. The study could be extended to include a larger sample size and a more comprehensive analysis of the behavior of matrix spectra associated with rational sequences. Future research could also involve the development of new methods for analyzing the properties of rational sequences.

    Coherent sequences of matrix spectra, draft 11aug26 v3 · 2026 · DOI

  • The paper does not provide a comprehensive comparison with existing methods. The approach is limited to unitary groups and may not be applicable to other compact groups.

    Weingarten Calculus with Virtual Isometries · 2026 · DOI

  • Defining multiplication on kaiso complex numbers. Analyzing the properties of kaiso complex numbers in different contexts.

    Noncommutative Infinite Kaiso · 2026 · DOI

  • The lack of a unified algebraic framework for kaiso complex numbers. The need for a closed formula for the inversion parity.

    Noncommutative Infinite Kaiso · 2026 · DOI

  • Further research is needed to explore the applications of the new approach to Weingarten calculus. The approach can be extended to other compact groups and used to study their properties. The results can be used to develop new methods for calculating moments of Haar-distributed unitary matrices.

    Weingarten Calculus with Virtual Isometries · 2026 · DOI

  • Further research is needed to fully understand the systematic cancellations coming from the symmetries of the matrices. The methods used in the paper may be applicable to other areas of mathematics. The paper's findings may be relevant to the study of symmetric functions and Hecke theory.

    Coherent sequences of matrix spectra, draft 7aug26 v3 · 2026 · DOI

  • Solving the Hermitian positive definite systems, which are sparse but ill-conditioned, involves using iterative methods, such as Conjugate Gradient (CG), which are time-consuming and computationally expensive.

    Matrix-free Neural Preconditioner for the Dirac Operator in Lattice Gauge Theory · 2026 · DOI

  • Further study of the properties of the Hilbert matrix. Application of the paper's techniques to other matrices. Exploration of the connections between the Hilbert matrix and other areas of mathematics.

    Tricks or Treats with the Hilbert Matrix · 1983 · DOI

  • A gap in the literature is identified, and the paper claims to contribute to the field.

    Hankel and Toeplitz matrices and forms: Algebraic theory · 1983 · DOI

  • The paper identifies a gap in the understanding of the Hilbert matrix. The paper aims to fill this gap by exploring new properties of the matrix.

    Tricks or Treats with the Hilbert Matrix · 1983 · DOI

  • In this work, we have illustrated how a class of NEPvs can be transformed into a NEP in a way that enables the use of NEP solvers. The transformation involves a polynomial system of equations that must be solved each time the NEP is accessed. We propose 123 BIT Numerical Mathematics (2026) 66:30 Page 23 of 26 30 Table 1 Performance metrics from the simulation. Each solve with the SMW formula involves six full-sized linear solves. One evaluation of G(λ) and H (λ) can be performed together using five linear solves. All linear solves are sparse.

    From eigenvector nonlinearities with quadratic structure to eigenvalue nonlinearities with algebraic structure · 2026 · DOI

  • 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

  • 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

  • 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

  • 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 instability of the linear systems resulting from the use of MQ-RBFs. The need for efficient and accurate methods for solving the 3D Helmholtz equation in complex geometries and three-dimensional problems.

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

  • The paper identifies a gap in the existing literature regarding the effectiveness of preconditioned LSQR with iterative refinement. 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

  • 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

  • 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

Most-cited papers in Matrix Theory and Algorithms

Most recent work

Find a gap in your own Matrix Theory and Algorithms sub-topic

This page shows what the Matrix Theory and Algorithms literature already flags as unresolved. To narrow it to your specific question, run the guided finder — it searches the gap library on demand and checks candidates against 250M+ OpenAlex works.

Open the Research Gap Finder →

Related topics in Computer Science

Online Learning and Analytics
851 open questions
Digital literacy in education
518 open questions
AI in Service Interactions
517 open questions
Music Technology and Sound Studies
423 open questions
Teaching and Learning Programming
384 open questions
Quantum Computing Algorithms and Architecture
368 open questions
Virtual Reality Applications and Impacts
317 open questions
Intelligent Tutoring Systems and Adaptive Learning
298 open questions
60 open questions have been extracted from the limitations and future-work passages of 557 Matrix Theory and Algorithms papers in our library. Each one below links back to the study that raised it, so you can read the original claim in context.

Tools for your next paper

Gaps & Literature SearchFind open questions in any field — 120,000+ open gaps mined from our own library.Pre-CheckIs your paper ready? A Tier 1–5 acceptance probability in seconds.Journal FinderA ranked shortlist of target journals from a 17,500-venue index.Duplicate Publication CheckerCheck prior-publication & salami-slicing risk across 8 sources.Citation GeneratorFreeA DOI or title → APA, MLA, Vancouver, Chicago, BibTeX, RIS. No signup.Graphical Abstract MakerFreeTurn your findings into a colorblind-safe graphical abstract.

Compare the category — Honest roundups of the AI research tools, ours listed alongside the alternatives.

Command palette

Jump anywhere, run any action.