1. Representations of Certain Matrices and Systems of Matrix Equations over Non-commutative Rings
Wang Qing-Wen
Depart. of Math.
Shanghai University
2019/4/25

2. Simultaneous decomposition of many matrices over any division rings
The past and current status of matrix theory over non-commutative rings
Simultaneous decomposition of many matrices over any division rings
Various symmetric matrices and generalized positive semidefinite quaternion matrices
Systems of matrix equations over regular rings
Systems of generalized Sylvester equations over polynomial rings over a finite central algebra
Some open problems
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3. 1. The past and current status of matrix theory over non-commutative rings
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4. In 1843, Hamilton（1805-1865）discovered the quaternion.
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5. Professor Hua and Wan once said that it is very valuable to study matrices over division rings
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6. Many famous mathematicians, such as Cohn, Jacobson, Thompson, Wiegman, Guralnick, Gustafson, Hua, Xie, Wan, and others, made great contributions to matrix theory over non-commutative rings.
The fundamental theory on reducing matrices by similarity established by P.M. Cohn.
The fundamental centralizable theory on matrices over skewfields established by Xie Bangjie.
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7. The main contents of matrices over division rings
Quaternion matrix theory
Eigenvalus and elementary factors of matrices
Generalized inverses of matrices
determinants
Matrix equations and systems of matrix equations
Matrix inequalities
Canonical forms and invariables of matrices
Matrix geometry.
除环上矩阵的研究也越来越引人注目, 已是目前矩阵论研究的活跃课题之一, 并成为代数学研究的一个新的生长点.
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8. The main two tasks in investigating matrix equations
The tools and methods
Solvable conditions, expressions of general solutions, and certain constraint solutions, such as centrosymmetric solution
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9. Main tools Decompositions of matrices Generalized inverses of matrices
Transformations of matrices
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10. 2. Equivalence canonical forms of matrix triplets over any division rings
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12. A Practical Algorithm
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16. Applications of equivalence canonical forms of matrix triplets
To investigate some linear matrix equations and systems over any division rings, such as
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17. To consider the independence of generalized Schur complement
To study the rank of a partitioned matrix, rank minimization of a partitioned matrix, and the connection with generalized Schur complement
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18. 4. Various symmetric matrices and generalized positive semidefinite matrices
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21. Generalized reflexive matrix
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22. Hsin-Chu Chen, Generalized reflexive matrices: special properties and applications, SIAM J. Matrix Anal. Appl. 19(1): 1998.
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23. Criteria for a matrix to be centrosymmetric and centroskewsymmetric
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26. Criteria for a matrix to be bisymmetric and biskewsymmetric
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28. Generalized positive semidefinite matrix
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31. 3. Systems of matrix equations over von Neumann regular ring
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32. An important system of matrix equations over a regular ring
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35. Various symmetric constant solutions to the system of generalized Sylvester matrix equations over a polynomial ring over a finite central algebra
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36. Sylvester matrix equation Generalized Sylvester matrix equation
System of generalized Sylvester matrix equations
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37. System of Sylvester matrix equations
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39. 6. Some open problems
How to generalize the Roth’s similarity theorem to an arbitrary division ring?
How to find out various symmetric solutions to the system of generalized Sylvester equations over a polynomial ring over a finite central algebra?
How to investigate nonlinear matrix equations over division rings and regular rings?
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40. Thanks!
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