last modified Jul 18, 1996

Our Non Commutative Algebra Package which runs under Mathematica© can be downloaded. If you have any trouble getting our software, then send us email at ncalg@osiris.ucsd.edu.

For an introduction to NCAlgebra and for elementary exercises read NCDOCUMENT.tex. It is available in TeX and Dvi formats (Oct 1994). Here is a short tutorial of some of the most basic commands. This site focuses on using the features of NCAlgebra to solve more advanced problems. This site contains several examples of problems which can be investigated with the aid of the features in NCAlgebra for discovering formulas. The goal is to test these methods on a variety of problems, most of which are classic theorems in some field.

Some of these problems have been solved, but several have not. Even those which have solutions posted lead to other unanswered questions. We would like this site to be as interactive as possible. If you have a solution to one of the problems posted, or you have another problem that you think our software could help solve, please contact us at ncalg@osiris.ucsd.edu, and we would be happy to post it here. If you have used our ``discovering methods'' to solve a problem, and you have your own work posted, let us know so we can refer to your page here.

SYSTEMS ENGINEERING

• Factoring systems
  • Minimal factorization (Theorem of Bart, Gohberg, Kaashoek, Van Dooren) Jun 96
    Solution by Helton, Schneider, Stankus and Yoshinoba
• Model reduction problems
  • Balanced truncation Jun 96
    Contributed by Keith Glover
    
    Solutions by Helton, Schneider and Stankus
• H-infinity control
  • Derivation of the X and Y Ricatti equations Jun 96
    Solution by Helton, Schneider, Stankus and Yoshinoba
• Orbit problems
  • Orbits of U(n,n) acting on Mn Jun 96
    Contributed by Nicholas Young
    
    Solution by Helton, Schneider and Stankus
• Other

OPERATOR THEORY

• Matrix completion problems
  • Invertible Matrix Completion Problem Jun 96
    Contributed by Hugo Woerdeman
    
    Solution by Helton, Schneider, Stankus and Yoshinoba

    Parrot matrix completion problem - the equality case Jun 96

    Solutions by Helton, Schneider and Stankus
• Other
  • Sum of Idempotents Jun 96
    Contributed by H. Bart and J. Stampfli
    
    Solutions by Stankus and Sloboda
  • H. Bart and J. Kaashoek problem (open)

PROBLEMS ORDERED ACCORDING TO METHODOLOGY REQUIRED TO SOLVE

• Gröbner basis only
  • Sum of Idempotents Jun 96
• 1-strategies
  • Invertible Matrix Completion Problem Jun 96
  • Parrot matrix completion problem - the equality case Jun 96
  • Minimal factorization Jun 96
  • Balanced truncation (parts a and b) Jun 96
  • Orbits of U(n,n) acting on Mn Jun 96
• 2-strategies
  • Derivation of the X and Y Ricatti equations (requires decompose) Jun 96
• Open
  • Balanced truncation (part c) Jun 96
  • H. Bart and J. Kaashoek problem