Cyclizable matrix pairs over image and a conjecture on Töplitz pencils Original Research Article

The 22-year-old conjecture is addressed which claims that any reachable matrix pair (A,B) from image is cyclizable, i.e. allows for F,u over image s.t. (A+BF)kBu, k=0,…,n−1, is a basis for image. It is shown that for a whole class of pairs the correctness of the conjecture is a consequence of the correctness of a conjecture on certain Töplitz pencils. An algebraic computable test for the validity of the latter is given. Based on these results the validity of the conjecture on cyclizability can be extended up to dimension 5.