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

Abstract

We investigate the use of noncommutative Groebner bases in solving partially prescribed matrix inverse completion problems. The types of problems considered here are similar to those in [BLJW]. There the authors gave necessary and sufficient conditions for the solution of a two by two block matrix completion problem. Our approach is quite different from theirs and relies on symbolic computer algebra. Here we describe a general method by which all block matrix completion problems of this type may be analyzed if sufficient computational power is available. We also demonstrate our method with an analysis of all three by three block matrix inverse completion problems with eleven blocks known and seven unknown. We discover 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 31,824 three by three block matrix completion problems with eleven blocks known and seven unknown. A solution to this problem of the form derived in [BLJW] is presented. Not only do we give a proof of our detailed result, but we describe the strategy used in discovering our theorem and proof, since it is somewhat unusual for these types of problems.