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

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

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• 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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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.

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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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Main tools

• Decompositions of matrices
• Generalized inverses of matrices
• Transformations of matrices

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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

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

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

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System of Sylvestermatrix equations
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6. Some open problems

• How to generalize the Roth