Representations of Certain Matrices and Systems of Matrix Equations over Non-commutative Rings Wang Qing-Wen Depart. of Math. Shanghai University 2019/4/25
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 2019/4/25
1. The past and current status of matrix theory over non-commutative rings 2019/4/25
In 1843, Hamilton（1805-1865）discovered the quaternion. 2019/4/25
Professor Hua and Wan once said that it is very valuable to study matrices over division rings 2019/4/25
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. 2019/4/25
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. 2019/4/25
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 2019/4/25
Main tools Decompositions of matrices Generalized inverses of matrices Transformations of matrices 2019/4/25
2. Equivalence canonical forms of matrix triplets over any division rings 2019/4/25
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A Practical Algorithm 2019/4/25
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Applications of equivalence canonical forms of matrix triplets To investigate some linear matrix equations and systems over any division rings, such as 2019/4/25
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 2019/4/25
4. Various symmetric matrices and generalized positive semidefinite matrices 2019/4/25
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Generalized reflexive matrix 2019/4/25
Hsin-Chu Chen, Generalized reflexive matrices: special properties and applications, SIAM J. Matrix Anal. Appl. 19(1): 1998. 2019/4/25
Criteria for a matrix to be centrosymmetric and centroskewsymmetric 2019/4/25
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Criteria for a matrix to be bisymmetric and biskewsymmetric 2019/4/25
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Generalized positive semidefinite matrix 2019/4/25
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3. Systems of matrix equations over von Neumann regular ring 2019/4/25
An important system of matrix equations over a regular ring 2019/4/25
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Various symmetric constant solutions to the system of generalized Sylvester matrix equations over a polynomial ring over a finite central algebra 2019/4/25
Sylvester matrix equation Generalized Sylvester matrix equation System of generalized Sylvester matrix equations 2019/4/25
System of Sylvester matrix equations 2019/4/25
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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? 2019/4/25
Thanks! 2019/4/25