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New derivations of classical theorems: idealized picture

We shall see in this paper that the algebra components of four theorems can be ``discovered'' using a strategy:
(1) The minimal factorization of a system due to Bart-Gohberg-Kaashoek and van Dooren.
(2) The Doyle-Glover-Khargonekar-Francis theorem of tex2html_wrap_inline4396 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 tex2html_wrap_inline4408 -decompositions which would consist of finding a polynomial k in tex2html_wrap_inline4408 unknowns and tex2html_wrap_inline4408 polynomials tex2html_wrap_inline4416 such that when tex2html_wrap_inline4418 is substituted for the j-th unknown for tex2html_wrap_inline4422 , one obtains p. An tex2html_wrap_inline4408 -Decompose operation would then be an operation which found all j-motivated unknowns for tex2html_wrap_inline4422 and an idealized tex2html_wrap_inline4408 -strategy would allow the use of the tex2html_wrap_inline4408 -Decompose operation.

A variant on 1-Decompose which we shall use frequently is called symmetric 1-Decompose. This applies in an algebra with involution, tex2html_wrap_inline4436 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 tex2html_wrap_inline4444 where the second polynomial tex2html_wrap_inline4446 is the adjoint of the first.gif

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.gif 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).

next up previous contents
Next: A practical picture Up: A highly idealized picture Previous: Idealized strategies

Wed Jul 3 10:27:42 PDT 1996