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Representations of Certain Matrices and Systems

of Matrix Equations over Non-commutative Rings

Wang Qing-Wen Depart. of Math. Shanghai

University

- 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

1. The past and current status of matrix theory

over non-commutative rings

In 1843, Hamilton(1805-1865)discovered the

quaternion.

- Professor Hua and Wan once said that it is very

valuable to study matrices over division rings

- 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.

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.

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

Main tools

- Decompositions of matrices
- Generalized inverses of matrices
- Transformations of matrices

2. Equivalence canonical forms of matrix triplets

over any division rings

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A Practical Algorithm

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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

- 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

4. Various symmetric matrices and generalized

positive semidefinite matrices

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Generalized reflexive matrix

- Hsin-Chu Chen, Generalized reflexive matrices

special properties and applications, SIAM J.

Matrix Anal. Appl. 19(1) 1998.

Criteria for a matrix to be centrosymmetric and

centroskewsymmetric

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- Criteria for a matrix to be bisymmetric and

biskewsymmetric

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Generalized positive semidefinite matrix

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3. Systems of matrix equations over von Neumann

regular ring

An important system of matrix equations over a

regular ring

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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

- Sylvester matrix equation
- Generalized Sylvester matrix equation
- System of generalized Sylvester matrix equations

System of Sylvestermatrix equations

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6. Some open problems

- How to generalize the Roth