  A word $w$ of letters on edges of underlying graph $\Gamma$ of deterministic
finite automaton (DFA) is called synchronizing if $w$ sends all states of the
automaton to a unique state. J. \v{C}erny discovered in 1964 a sequence of
$n$-state complete DFA possessing a minimal synchronizing word of length
$(n-1)^2$. The hypothesis, well known today as \v{C}erny conjecture, claims
that $(n-1)^2$ is a precise upper bound on the length of such a word over
alphabet $\Sigma$ of letters on edges of $\Gamma$ for every complete $n$-state
DFA. The hypothesis was formulated distinctly in 1966 by Starke. A special
classes of matrices induced by words in the alphabet of labels on edges of the
underlying graph of DFA are used for the study of synchronizing automata.
