Matrix cyclization over complex polynomials Original Research Article

Let A be an n × n, B an n × m complex polynomial matrix. The following open problem is investigated: does there exist a complex polynomial (m × n)-matrixF such that A + BF has a cyclic vector in the image of B? An explicit solution of the problem is given for the following generic situation: n greater-or-equal, slanted 4, [B,AB,…,An−1B] is rightinvertible (necessary), the entries of a specific normalized form of (A, B) do not satisfy certain polynomial equations, and one specific entry has at least one simple zero outside a certain finite set. The exceptional equations are given explicitely and can easily be checked for.