Row monomial matrices and Černy conjecture, short proof

The class of row monomial matrices (one unit and rest zeros in every row) with some non-standard operations of summation and usual multiplication is our main object. These matrices generate a space with respect to the mentioned operations. 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 ($|R(w)|=1$. 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 the \v{C}erny conjecture, claims that $(n-1)^2$ is also precise upper bound on the length of such a word for a complete DFA. The hypothesis was formulated in 1966 by Starke. The problem has motivated great and constantly growing number of investigations and generalizations. The proof of the conjecture is based on connection between length of words $u$ and dimension of the space generated by row monomial matrices $M_u$, the set of synchronizing matrices placed some role.