Cyclic dimensions, controllability subspaces and Gohberg-Kaashoek numbers (Q1893859)

Two alternative proofs of a theorem of \textit{V. Macaev} and \textit{V. Olshevsky} [Cyclic dimensions, kernel multiplicities and Gohberg-Kaashoek numbers, Linear Algebra Appl. (to appear)]\ are given, in a system theory approach. The \(d\)-th cyclic dimension \(r_d (A)\) of an \(n\times n\) matrix \(A\) is defined by \(r_d (A)= \max \{\dim S_A (f_1, \dots, f_d):\) \(f_1, \dots, f_d\in \mathbb{C}^n \}\), \(d=1, \dots, n\), where \[ S_A (f_1, \dots, f_d)= \text{sp} \{f_1, \dots, f_d, Af_1, \dots, Af_d, \dots, A^{n-1} f_1, \dots, A^{n-1} f_d\}. \] It is proved that \(r_d (A)= \sum_{i=1}^d \deg p_i\), where \(p_n |p_{n-1} |\cdots |p_2 |p_1\) are the invariant polynomials of \(A\). The cyclic dimension is redefined in terms of the controllability space of a pair \((A, B)\) which is \(\text{Im} (A, B)= \text{Im} [B\;AB\dots A^{n-1} ]\), hence \(r_d (A)= \max\{ \dim \text{ Im} (A, B)\mid B\in \mathbb{C}^{n\times d}\}\). Using the first natural form with companion blocks of \(A\), the canonical form and the controllability indices of a pair \((A, B)\), two proofs are obtained, as well as a dual result concerning the relationship between the Gohberg-Kaashoek numbers \(\deg p_i\) and the dimension of the unobservable space of the pair \((C, A)\).