Archive - AMS Classification 15A18

Let \(F\) be an infinite field and let \(n,p_{1},p_{2},p_{3}\) be positive integers such that \(n=p_{1}+p_{2}+p_{3}.\) Let \[(C_{1},C_{2})=\left( \begin{bmatrix} C_{1,1} & C_{1,2} \\ C_{2,1} & C_{2,2} \end{bmatrix} , \begin{bmatrix} C_{1,3} \\ C_{2,3} \end{bmatrix} \right) ,\] where the blocks \(C_{i,j}\) are of type \(p_{i}\times p_{j},i\in \{1,2\},j\in \{1,2,3\}.\) We analyse the possibility of the pair \((C_{1},C_{2})\) being completely controllable, when

(i) \(C_{1,2},\) \(C_{1,3}\), and \(C_{2,1}\) are fixed and the other blocks vary;

(ii) \(C_{1,1},\) \(C_{1,2}\), and \(C_{2,1}\) are fixed and the other blocks vary.

We still describe the possible characteristic polynomials of a partitioned matrix of the form \(C=[ C_{i,j}] \in F^{n\times n},\) where the blocks \(C_{i,j}\) are of type \(p_{i}\times p_{j},i,j\in \{1,2,3\}\), when one of the conditions (i) or (ii) occurs.

Soit \(F\) un corps infini et soient \( n,p_{1},p_{2},p_{3}\) des entiers positifs tels que \(n=p_{1}+p_{2}+p_{3}.\) Soit \[(C_{1},C_{2})=\left( \begin{bmatrix} C_{1,1} & C_{1,2} \\ C_{2,1} & C_{2,2} \end{bmatrix} , \begin{bmatrix} C_{1,3} \\ C_{2,3} \end{bmatrix} \right) ,\] où les blocs \(C_{i,j}\) sont de type \(p_{i}\times p_{j},i\in \{1,2\},j\in \{1,2,3\}.\) Nous établions conditions pour lesquelles \((C_{1},C_{2})\) est controllable, quand

(i) \(C_{1,2},C_{1,3}\), et \(C_{2,1}\) sont connus et les autres blocs varient;

(ii) \(C_{1,1},C_{1,2}\), et \(C_{2,1}\) sont connus et les autres blocs varient.

Soit \(C=[ C_{i,j}] \in F^{n\times n},\) où les blocs \(C_{i,j}\) sont de type \(p_{i}\times p_{j},i,j\in \{1,2,3\}.\) Nous étudions le polynôme caractéristique de la matrice \(C,\) quand une des conditions (i) ou (ii) est satisfait.