Solving the HGRAIL using NCProcess: Converse

The straightforward converse of what we have just prooved would be: If items (1), (2) and (3) hold, then (HGRAIL) has a solution with invertible and b and c are given by the formulas in Step 2. There is no reason to believe (and it is not the case) that b and c must be given by the formulas in Step 2. These two formulas were motivated by ``excess freedom'' in the problem. The converse which we will attempt to prove is:

Proposed Converse If items (1), (2) and (3) hold, then (HGRAIL) has a solution with invertible.

To obtain this proposed converse, we need a complete spreadsheet corresponding to the last stages of our analysis.

In the spreadsheet, we use conventional , notation rather than ``discovered'' notation so that our arguments will be familiar to experts in the field of control theory.

Now we use the above spreadsheet to verify the proposed converse. To do this, we assume that matrices , , , , and exist, that and are invertible, that and are self-adjoint, that , is invertible and that the DGKF and equations hold. That is, the two following polynomial equations hold.

We wish to assign values for the matrices , , , , , and such that each of the equations on the above spreadsheet hold. If we can do this, then each of the equations from the starting polynomial equations will hold and the proposed converse will follow.

  1. Note that all of the equations in the {}-Category of the above spreadsheet hold since X and Y solve the DGKF equations and are both invertible.
  2. Set equal to the inverse of Y. This assignment is dictated by the user selects. Note that

    follows since Y is self-adjoint.
  3. Let and be any invertible matrices such that . For example, one could choose and to both be the identity matrix.
  4. Note that there is there is a user select
    and that and are invertible. Since is invertible and , is invertible. Therefore, we set
    .
    Since , and , it follows that is invertible and self-adjoint.
  5. Since has been set for i, j = 1, 2, we can set a, b and c according to their formulas at the top of the spreadsheet.

With the assignments of , , , , a, b and c as above, it is easy to verify by inspection that every polynomial equation on the spreadsheet above holds.

We have proven the proposed converse and, therefore, have proven the following approximation to the classical [DGKF] theorem.

Theorem If (HGRAIL) has a solution with invertible and b and c are given by the formulas from step 2, then the DGKF and equations have solutions and which are symmetric matrices with , and invertible. If the DGKF and equations have solutions and which are symmetric matrices with , and invertible, then (HGRAIL) has a solution with invertible .

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