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The Optimal Version of Hua’s Fundamental Theorem of Geometry of Rectangular Matrices
 
Peter Šemrl University of Ljubljana, Ljubljana, Slovenia
The Optimal Version of Hua's Fundamental Theorem of Geometry of Rectangular Matrices
eBook ISBN:  978-1-4704-1892-2
Product Code:  MEMO/232/1089.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $39.00
The Optimal Version of Hua's Fundamental Theorem of Geometry of Rectangular Matrices
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The Optimal Version of Hua’s Fundamental Theorem of Geometry of Rectangular Matrices
Peter Šemrl University of Ljubljana, Ljubljana, Slovenia
eBook ISBN:  978-1-4704-1892-2
Product Code:  MEMO/232/1089.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $39.00
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2322014; 74 pp
    MSC: Primary 15; 51;

    Hua's fundamental theorem of geometry of matrices describes the general form of bijective maps on the space of all \(m\times n\) matrices over a division ring \(\mathbb{D}\) which preserve adjacency in both directions. Motivated by several applications the author studies a long standing open problem of possible improvements.

    There are three natural questions. Can we replace the assumption of preserving adjacency in both directions by the weaker assumption of preserving adjacency in one direction only and still get the same conclusion? Can we relax the bijectivity assumption? Can we obtain an analogous result for maps acting between the spaces of rectangular matrices of different sizes?

    A division ring is said to be EAS if it is not isomorphic to any proper subring. For matrices over EAS division rings the author solves all three problems simultaneously, thus obtaining the optimal version of Hua's theorem. In the case of general division rings he gets such an optimal result only for square matrices and gives examples showing that it cannot be extended to the non-square case.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • 2. Notation and basic definitions
    • 3. Examples
    • 4. Statement of main results
    • 5. Proofs
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2322014; 74 pp
MSC: Primary 15; 51;

Hua's fundamental theorem of geometry of matrices describes the general form of bijective maps on the space of all \(m\times n\) matrices over a division ring \(\mathbb{D}\) which preserve adjacency in both directions. Motivated by several applications the author studies a long standing open problem of possible improvements.

There are three natural questions. Can we replace the assumption of preserving adjacency in both directions by the weaker assumption of preserving adjacency in one direction only and still get the same conclusion? Can we relax the bijectivity assumption? Can we obtain an analogous result for maps acting between the spaces of rectangular matrices of different sizes?

A division ring is said to be EAS if it is not isomorphic to any proper subring. For matrices over EAS division rings the author solves all three problems simultaneously, thus obtaining the optimal version of Hua's theorem. In the case of general division rings he gets such an optimal result only for square matrices and gives examples showing that it cannot be extended to the non-square case.

  • Chapters
  • 1. Introduction
  • 2. Notation and basic definitions
  • 3. Examples
  • 4. Statement of main results
  • 5. Proofs
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.