**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.

- 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. - Set
equal to the
inverse of
*Y*. This assignment is dictated by the user selects. Note that

follows since*Y*is self-adjoint.

- Let
and
be any invertible
matrices such that
.
For example, one could
choose
and
to both be the identity matrix.

- Note that there is
there is a user select
. - 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
.
*