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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 ourmain object. These matrices generate a space with respect to the mentionedoperations. A word w of letters on edges of underlying graph of deterministicfinite automaton (DFA) is called synchronizing if w sends all states of theautomaton to a unique state J. \v{C}erny discovered in 1964 a sequence ofn-state complete DFA possessing a minimal synchronizing word of length(n-1)(n-1). The hypothesis, well known today as the \v{C}erny conjecture,claims that (n-1)(n-1) is also precise upper bound on the length of such a wordfor a complete DFA. The hypothesis was formulated in 1966 by Starke. Theproblem has motivated great and constantly growing number of investigations andgeneralizations. We present the proof of the \v{C}erny-Starke conjecture: thedeterministic complete n-state synchronizing automaton has synchronizing wordof length at most (n-1)(n-1). The proof used connection between dimension ofthe space and the length of words on paths of edges in underlying graph ofautomaton.

Row monomial matrices and \v{C}erny conjecture, short proof