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By Prof. Bruce A. Francis (eds.)

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Defining K:=G[ 0 we get Thus the definition gives the desired result, that is inverfible in R t L . [] The obvious dual of Lemma 2 is that M and N are left-coprime iff there exist U and V such that Proof of Theorem 1. We shall prove equivalence of (i) and (ii). 1 there exist U and V in RH~ such that [0 I ] N V ~ RH~. Ch. 4 32 This implies by Lemma 2 and its dual that M, [0 I]N are right-coprime and ro] M, [iJ U are left-coprime. But the latter condition implies left-coprimeness of i,0] (ii) ~ (i): Choose, by right-coprimeness and Lemma 2, matrices X and Y in RHo~ such that M X] -1 [0 I]N ~ RH~,.

Exercise 1. Verify that the matrices in (6) and (7) satisfy (3). Example 1. As an illustration of the use of these formulas, consider the scalar-valued example G (s) = s-1 s (s-2) A minimal realization is G(s)=[A,B, C,D] c=[-1 11, D = 0 . Choosing F to place the eigenvalues Of AF (arbitrarily) at { - 1 , - 1 }, we get f = [-1 -4] Then N(s)=[AF, B, C, O] s-1 = (s+i)2 and M(s) = [AF, B, F, 1] s (s -2) (S +1)2 " Similarly, the assignment Ch. 4 25 yields X (s) = [AF, - n , c , 11 ff+6s-23 (s+l) 2 Y ( s ) = [ A F , - 1 t , F, O] -41s+l (s+l)2 " Finally, in this example we have N = N , aT/=M, :~=X, Y=Y.

4 equivalently, [F G ] is right-invertible in RH~. Now let G be a proper real-rational matrix. A right-coprimefactorization of G is a factorization G--NM -1 where N and M are right-coprime RH,~-matrices. Similarly, a left-coprime factorization has the form G=~I-I~[ where/q and/~ are left-coprime. Of course implicit in these definitions is the requirement that M and ~ / b e square and nonsingular. We shall require special coprime factorizations, as described in the next lemma. Lemma 1. For each proper real-rational matrix G there exist eight RH~-matrices satisfying the equations G =NM -1 =AT/-iN (2) Equations (2) and (3) together constitute a doubly-coprime factorization of G.

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