Mathematical Reports - Comptes rendus mathématiques
of the Academy of Science | de l'Académie des sciences
The averaged joint spectral radius (AJSR) is defined. By using the avalanche principle we develop an effective algorithm to compute the averaged joint spectral radius for a pair of \(2\times2\) matrices.
Nous introduisons la notion de rayon spectral moyen d’un ensemble fini de matrices. En utilisant le principe d’avalanche, nous développons un algorithme efficace pour calculer le rayon spectral moyen d’une paire de matrices de tailles \(2\times2\).
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.
