Using noncommutative Grobner bases in solving partially prescribed matrix inverse completion problems

We investigate the use of noncommutative Grobner bases in solving partially prescribed matrix inverse completion problems. The types of problems consi dered here are similar to those in [Linear Algebra Appl. 223-224 (1995) 73] . There the authors gave necessary and sufficient conditions for the soluti on of a 2 x 2 block matrix completion problem. Our approach is quite differ ent from theirs and relies on symbolic computer algebra. Here we describe a general method by which all block matrix completion prob lems of this type may be analyzed if sufficient computational power is avai lable. We also demonstrate our method with an analysis of all 3 x 3 block m atrix inverse completion problems with I I blocks known and 7 unknown. We d iscover that the solutions to all such problems are of a relatively simple form. We then perform a more detailed analysis of a particular problem from the 3 1,824 3 x 3 block matrix completion problems with I I blocks known and 7 un known. A solution to this problem of the form derived in the above-mentione d reference is presented. Not only do we give a proof of our detailed result, but we describe the str ategy used in discovering our theorem and proof, since it is somewhat unusu al for these types of problems. (C) 2001 Elsevier Science Inc. All rights r eserved.