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Review

A Review on Solving Sylvester-Type Equations

1
Department of Mathematics, Shanghai University, Shanghai 200444, China
2
Collaborative Innovation Center for the Marine Artificial Intelligence, Shanghai University, Shanghai 200444, China
*
Author to whom correspondence should be addressed.
Symmetry 2026, 18(6), 984; https://doi.org/10.3390/sym18060984
Submission received: 6 May 2026 / Revised: 26 May 2026 / Accepted: 29 May 2026 / Published: 6 June 2026
(This article belongs to the Special Issue Mathematics: Feature Papers 2026)

Abstract

The solution theory of Sylvester-type equations finds wide applications in control theory, robotics, and image processing. This paper systematically surveys, classifies and summarizes the existing research results of three classes of Sylvester-type equations: matrix equations, tensor equations, and operator equations. It extracts nine mainstream research methods and clarifies the internal correlations among these methods, as well as their applicable equation types. Combined with four prior review articles focusing on special cases of Sylvester-type equations, this work establishes a comprehensive framework for solving such equations. It not only provides a systematic theoretical foundation and a clear research thread for subsequent researchers but also offers valuable methodological insights for further investigations in related fields.
Keywords: Sylvester-type equations; Sylvester-type matrix equations; Sylvester-type tensor equations; Sylvester-type operator equations; singular value decomposition; generalized singular value decomposition; canonical correlation decomposition; product singular value decomposition; simultaneous decomposition; equivalent decomposition; generalized inverses; rank equations; extremal ranks; real representations; complex representations; Kronecker product; vectorization operator; L-structures; \({\mathcal{H}}\)-representations; determinantal representations; semi-tensor product Sylvester-type equations; Sylvester-type matrix equations; Sylvester-type tensor equations; Sylvester-type operator equations; singular value decomposition; generalized singular value decomposition; canonical correlation decomposition; product singular value decomposition; simultaneous decomposition; equivalent decomposition; generalized inverses; rank equations; extremal ranks; real representations; complex representations; Kronecker product; vectorization operator; L-structures; \({\mathcal{H}}\)-representations; determinantal representations; semi-tensor product

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MDPI and ACS Style

Wang, Q.-W.; Gao, J. A Review on Solving Sylvester-Type Equations. Symmetry 2026, 18, 984. https://doi.org/10.3390/sym18060984

AMA Style

Wang Q-W, Gao J. A Review on Solving Sylvester-Type Equations. Symmetry. 2026; 18(6):984. https://doi.org/10.3390/sym18060984

Chicago/Turabian Style

Wang, Qing-Wen, and Jiale Gao. 2026. "A Review on Solving Sylvester-Type Equations" Symmetry 18, no. 6: 984. https://doi.org/10.3390/sym18060984

APA Style

Wang, Q.-W., & Gao, J. (2026). A Review on Solving Sylvester-Type Equations. Symmetry, 18(6), 984. https://doi.org/10.3390/sym18060984

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