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.
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 .