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Representations of Certain Matrices and Systems of Matrix Equations over Non-commutative Rings

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Title: Representations of Certain Matrices and Systems of Matrix Equations over Non-commutative Rings


1
Representations of Certain Matrices and Systems
of Matrix Equations over Non-commutative Rings
Wang Qing-Wen Depart. of Math. Shanghai
University
2
  • 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

3
1. The past and current status of matrix theory
over non-commutative rings
4
In 1843, Hamilton(1805-1865)discovered the
quaternion.
5
  • Professor Hua and Wan once said that it is very
    valuable to study matrices over division rings

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.

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.

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

9
Main tools
  • Decompositions of matrices
  • Generalized inverses of matrices
  • Transformations of matrices

10
2. Equivalence canonical forms of matrix triplets
over any division rings
11
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12
A Practical Algorithm
13
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14
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15
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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

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

18
4. Various symmetric matrices and generalized
positive semidefinite matrices
19
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21
Generalized reflexive matrix
22
  • Hsin-Chu Chen, Generalized reflexive matrices
    special properties and applications, SIAM J.
    Matrix Anal. Appl. 19(1) 1998.

23
Criteria for a matrix to be centrosymmetric and
centroskewsymmetric
24
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26
  • Criteria for a matrix to be bisymmetric and
    biskewsymmetric

27
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28
Generalized positive semidefinite matrix
29
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30
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31
3. Systems of matrix equations over von Neumann
regular ring
32
An important system of matrix equations over a
regular ring
33
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34
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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
36
  • Sylvester matrix equation
  • Generalized Sylvester matrix equation
  • System of generalized Sylvester matrix equations

37
System of Sylvestermatrix equations
38
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39
6. Some open problems
  • How to generalize the Roth
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