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OPERATOR THEORY

• Matrix completion problems
• 2 x 2 Invertible Matrix Completion Problem Jun 96
Contributed by Hugo Woerdeman
Solution by Helton, Schneider, Stankus and Yoshinobu
• 3 x 3 Invertible Matrix Completion Problem; 31,000 New Theorems April 1999
Solutions by Kronewitter
• Parrot matrix completion problem - the equality case Jun 96
Solutions by Helton, Schneider and Stankus
• Pseudoinverses : Gröebner Bases for basic situations
• Some notes(.dvi) (.tex) on the runs below which give finite and, in one case, an infinite Gröebner Basis for situations involving pseudoinverses.
• PseudoInverse of A has a finite GB. (.dvi) (.tex)
• PseudoInverse of A,B, and A + B has a finite GB. (.dvi) (.tex)
• PseudoInverse of A,B, and AB does NOT have a finite GB. (Four recursion laws.)
  dvi: (3 iters) (4 iters) (5 iters)
  tex: (3 iters) (4 iters) (5 iters)
• PseudoInverse of A,B, and A**Pinv[A]+B**Pinv[B] has a finite GB. (.dvi) (.tex)
• PseudoInverse of A,B, and A**Pinv[A] + Pinv[B]**B has a finite GB. (.dvi) (.tex)
• PseudoInverse of A,B, and (1-Pinv[B]**B)**A**Pinv[A] has a finite GB. (.dvi) (.tex)
• PseudoInverse of A,B, and (1-Pinv[B]**B)**A**Pinv[A] AND A**Pinv[A]+Pinv[B]**B has a finite GB. (.dvi) (.tex)
• Here is the mathematica file with all of the previous runs.
• Pseudoinverse of a column matrix.
• Here is the mathematica file for the pseudoinverse of a column matrix.
• Here is the (.dvi) (.tex) file for its output.
  Note that we are solving for the pseudoinverse of a 2x1 matrix with entries a and b where the pseudoinverse is a 1x2 matrix with entries c and d. It turns out that c = Pinv[tp[a]**a+tp[b]**b]**tp[a] and d = Pinv[tp[a]**a+tp[b]**b]**tp[b].
• 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)
• A solution to Yongge Tian's EP Matrices problem proposed in Image April 2001 volume 26.