Note that it is necessary that all of the equations in the spreadsheet have solutions, since they are implied by the original equations. The equations involving only knowns play a key role. In particular, they say precisely that, there must exist a projection such that
are satisfied.
The converse is also true and can be verified with the assistance of the above spreadsheet. To do this, we assume that the matrices A, B, C and are given and that (theTwoEquations) holds, and wish to define , , , , a, b, c, e, f and g such that each of the equations in the above spreadsheet hold. If we can do this, then each of the equations from the starting polynomial equations (FAC) given in §subsection:problem will hold and we will have shown that a minimal factorization of the [A,B,C,1] system exists.
With the assignments made above, every equation in the spreadsheet above holds. Thus, by backsolving through the spreadsheet, we have constructed the factors of the original system [A,B,C,1]. This proves
Theorem ([BGKvD]) The system [A, B, C, 1] can be factored if and only if there exists a projection such that and .
Note that the known equations can be neatly expressed in terms of and . Indeed, it is easy to check with a little algebra that these are equivalent to (eq:answer). It is a question of taste, not algebra, as to which form one chooses.
For a more complicated example of an end game, see §.