We consider block matrix completion problems similar to those in a paper in 1995 by W.W. Barrett, C.R. Johnson, M.E. Lundquist and H.J. Woerdeman, "Completing a Block Diagonal Matrix With a Partially Prescribed Inverse".
Here we take two partially prescribed, square matrices, A and B, and describe conditions which make it possible to complete the matrices so that they are inverses of each other. That is, we wish the completed matrices to satisfy
The particular block matrix inverse completion problems we have studied are those which have 7 unknown blocks and 11 known blocks. An example of which is
where the k's are known blocks and the u's are unknown.
Instead of looking at all of these problems we have only chosen to investigate one problem from the equivalence class of problems which are of the form
where Pi and Phi are permutation matrices. All of the problems with 7 unknown blocks and 11 known blocks have answers which are compressed in the file below, thms.tar.Z
They are indexed by some special scheme. The Mathematica notebook below will assist you in finding the Theorem number (index) of the particular 3x3 matrix inverse completion problem you may be interested in.
We investigate the use of noncommutative Groebner computer algebra
in solving partially prescribed matrix inverse completion
problems. The type of problems considered here are
similar to those in a paper in 1995 by W.W. Barrett, C.R. Johnson,
M.E. Lundquist and H.J. Woerdeman,
Here we describe a general method by which all block matrix completion problems of this type may be analyzed with sufficient computational power. We also demonstrate our method with an analysis of all three by three block matrix inverse completion problems with eleven known blocks and seven unknown. We discover that the solutions to all such problems are of a relatively simple form.
We then do a more detailed analysis of what can be considered the ``central problem" of the 31,824 three by three block matrix completion problems with eleven known blocks 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.
The Appendices in the document referenced above are