1 Controllability of Matrices with Prescribed Blocks

By Glória Cravo.

Let F be a field and let n, p_{1}, …, p_{k} be positive integers such that n = p_{1} + ⋯ + p_{k}. Let

(C_{1}, C_{2}) = ([ \begin{matrix}C_{1,1} & ⋯ & C_{1,k−1} \\ ⋮ & & ⋮ \\ C_{k−1,1} & ⋯ & C_{k−1,k−1} \end{matrix} ] , [ C_{1,k} ; ⋮ ; C_{k−1,k} ])

where the blocks C_{i,j} are of type p_{i} ×p_{j}, i ∈ {1, …, k −1}, j ∈ {1, …, k}. We study the possibility of (C_{1}, C_{2}) being completely controllable, when some of its blocks are fixed and the others vary. Our main results analyse the following cases: (i) All the blocks C_{i,j} are of the same size; (ii) The blocks C_{i,j} are not necessarily of the same size and k = 3. We also describe the possible characteristic polynomial of a matrix of the form

C = [ \begin{matrix}C_{1,1} & ⋯ & C_{1,k} \\ ⋮ & & ⋮ \\ C_{k,1} & ⋯ & C_{k,k} \end{matrix} ]

when some of its blocks are prescribed and the others are free.