- (1) The minimal factorization of a system due to Bart-Gohberg-Kaashoek and van Dooren.
- (2) The Doyle-Glover-Khargonekar-Francis theorem of control.
- (3) A matrix completion theorem due to W.W. Barrett, C.R. Johnson, M. E. Lundquist and H. Woerderman.
- (4) A matrix completion theorem due to Steve Parrot.

We can interpret this from a traditional viewpoint as the following theorem.

**Theorem 1.2***
The key formulas in Theorems 1,3,4 above can be derived with an
idealized strategy.
*

We use the name 1-Decompose to also refer
to the Decompose operation,
we use the name 1-motivated unknown to
refer to motivated unknown and we use the name
idealized 1-strategy to refer to an idealized strategy.
For the 1-Decompose operation,
the key was to find a polynomial *k* in
the knowns and
one unknown
and a polynomial *q* such that when *q* is substituted
for the one unknown of *k*, one obtains *p*.
Likewise, we could consider -decompositions
which would consist of finding a polynomial *k*
in unknowns and polynomials
such that when is substituted
for the *j*-th unknown for , one
obtains *p*. An -Decompose operation would then
be an operation which found all *j*-motivated unknowns
for and an idealized -strategy would allow
the use of the -Decompose operation.

A variant on 1-Decompose which we shall use
frequently is called *symmetric 1-Decompose*.
This applies in an algebra with involution, for
all *w*,
for example, a matrix algebra with adjoints or transposes.
Symmetric
1-Decompose applied to *p* yields
a 2-decomposition of *p* as
where the second polynomial
is the adjoint of the first.

We were not able to derive the key formulas of the theorem of item 2 above with a 1-strategy, but they can be derived with a symmetrized 2-strategy. The use of a symmetrized 2-strategy forces human intervention, but it is small because the 2-decomposition required is easy to recognize from the output of the NCProcess1 command. (Note that it originally took 5 years to discover the theorem).

Wed Jul 3 10:27:42 PDT 1996