Cyclizable matrix pairs over \(\mathbb C[x]\) and a conjecture on Toeplitz pencils
Given nonnegative integers \(n,m\) and a commutative ring \(R\), a matrix pair \((A,B)\) with \(A\in R^{n\times n}\), \(B\in R^{n\times m}\) is called reachable if \([B,AB,\dots,A^{n-1}B]\) is right invertible. The cyclization property claims that any reachable matrix pair is cyclizable, i.e., there exist \(u\in R^ m\), \(F\in R^{n\times m}\) such that \([Bu, (A+BF)Bu,\dots, (A+BF)^{n-1}Bu]\) is invertible. It is shown that for a whole class of pairs over \({\mathbb C}[x]\) the cyclization property holds if a certain conjecture on Toeplitz pencils of matrices over \({\mathbb C}[x]\) is fulfilled. An algebraic computable test for the validity of the latter is given. Based on these results the validity of the conjecture on cyclizability is show to be true up to dimension \(n=5\).
scientific article; zbMATH DE number 2117280