A NOTE ON LINEAR TRANSFORMATIONS WHICH LEAVE CONTROLLABLE MULTIINPUT DESCRIPTOR SYSTEMS CONTROLLABLE

Consider a generalized linear dynamical system Ex = Ax + Bu, where x i s an element of C-n, u is an element of C-m, and E, A, B are matrices of appropriate sizes with entries in C. This system, or the matrix tri ple (E, A, B), is called controllable if det(alpha E - beta A) is not a zero polynomial in alpha, beta and (alpha E - beta A, B) is of full rank for all (alpha, beta) is an element of C\{(0, 0)}. Let f be a lin ear transformation on C-nxn x C-nxm, the linear space of all matrix pa irs (A, B). In an earlier paper, Mehrmann and Krause attempted to prov e that, if f is of the form X --> UXV, and rank f(alpha E - beta A, B) = n for all (alpha, beta) is an element of C-2\{(0, 0)} and all contr ollable systems (E, A, B), then U, V are nonsingular matrix with V in some lower block triangular form. In this paper, we correct an error c ontained in this result and discuss whether the corrected result can b e generalized in such a way that no restrictions are placed on the for m of f.