We then run NCProcess once again, and we have yet another spreadsheet in which a, b, c and d are solved for. There are quite a few undigested relations in this spreadsheet to sort through. In order to eliminate them quickly we use SmallBasis. We will call SmallBasis in a slightly different way than we did in step 2. The most general call to SmallBasis looks like this.
Where list1 is a list of the relations which we wish to eliminate, and list2 is a list of the relations which we like. We hope that the relations in list2 generate the relations in list1. If this is the case, then the relations in list1 are redundant, and we can eliminate them. The SmallBasis command will return a subset of list1 which together with list2 generates the same ideal as the union of list1 and list2. We need to sort through the result of the previous NCProcess command to see which relations we want to put into which list.
We will construct the second list first. These are the relations which we know are true, or we are willing to assume are true. This includes all of the inverse relations. That is, a a - 1, a a - 1 and so forth. Also, the two intertwining relations go into list2. There is one with f and one with F. The second list also includes the equations which solve for a, b, c and d. In addition to these four there are four more relations which we can add to the set of digested relations. Since we have solved for a, b, c and d, we can take the transposes of these equations to solve for a, b, c and d. This is done with the following command.
TransposeRelations[stuff] |
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Where stuff can be any relation or list of relations including polynomials and rules. This command applies the NCAlgebra transpose function to stuff, and returns it. In this case, stuff is a list of the four rules which solve for a, b, c and d.
The first list in the call to SmallBasis is everything that we did not put into the second list. These equations will be eliminated if they can be generated by the polynomials in list1. If they can not be eliminated, then they could imply some sort of compatibility relation. This could mean that there is some other assumption that needs to be made in order for the constructions of a, b, c and d to produce the desired linear fractional transformation. We will see what to do with these relations after we see which ones survive the SmallBasis command.
The spreadsheet shows that the three digested relations above have been eliminated by SmallBasis. The expressions which solved for the indeterminates like a, b and c have also been removed. All that is left is the equations for the unknowns and their transposes as well as the inverse relations and the intertwinings of f and F. This leads to the following theorem, which is the converse to the theorem which we proved earlier.
Theorem: Suppose x, y and (1 - x x) are invertible matrices and there exists a matrix f and an invertible matrix F. such that
(1) | |
(2) |