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\begin{document}

\begin{center}
{\large Controllability of Matrices with Prescribed Blocks}

\bigskip

Gl\'{o}ria Cravo
\end{center}

Departamento de Matem\'{a}tica e Engenharias,

Universidade da Madeira

Portugal

Email address: gcravo@uma.pt

\begin{center}
\bigskip

{\Large Abstract}
\end{center}

\bigskip

\bigskip

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.
\end{document}
