A note on linear transformations which leave controllable multiinput descriptor systems controllable Original Research Article

Consider a generalized linear dynamical system image, where x set membership, variant Cn, uset membership, variant Cm, and E, A, B are matrices of appropriate sizes with entries in C. This system, or the matrix triple (E, A, B), is called controllable if det(αE − βA) is not a zero polynomial in α, β and (αE − βA, B) is of full rank for all (α, β) set membership, variant C (0, 0). Letf be a linear transformation on Cn×n × Cn×m, the linear space of all matrix pairs (A, B). In an earlier paper, Mehrmann and Krause attempted to prove that, if f is of the form X at UXV, and rank f(αE − βA, B) = n for all (α, β) set membership, variant C2 (0, 0) and all controllable 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 contained in this result and discuss whether the corrected result can be generalized in such a way that no restrictions are placed on the form of f.