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\section{Controllability of Matrices with Prescribed Blocks} By {\sl Gl\'{o}ria Cravo}.  

\medskip  \noindent Let $F$ be a field and let $n,p_{1},\ldots,p_{k}$ be positive integers such that $n=p_{1}+\cdots+p_{k}.$ Let \[ (C_{1},C_{2})=\left(  \left[ \begin{array} [c]{ccc} C_{1,1} & \cdots &  C_{1,k-1}\\ \vdots &  & \vdots\\ C_{k-1,1} & \cdots &  C_{k-1,k-1} \end{array} \right]  ,\left[ \begin{array} [c]{c} C_{1,k}\\ \vdots\\ C_{k-1,k} \end{array} \right]  \right) \] where the blocks $C_{i,j}$ are of type $p_{i}\times p_{j},i\in\{1,\ldots ,k-1\},j\in\{1,\ldots,k\}.$ 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 \[ C=\left[ \begin{array} [c]{ccc} C_{1,1} & \cdots &  C_{1,k}\\ \vdots &  & \vdots\\ C_{k,1} & \cdots &  C_{k,k} \end{array} \right] \] when some of its blocks are prescribed and the others are free.
%Controllability, Characteristic Polynomials, Matrix Completion Problems","93B05","15A18","","05:36:16","Sun Mar 02 2008","193.136.232.62"
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