  For a family of $n*n$ left triangular matrices with binary entries we derive
asymptotically exact (as $n\to\infty$) representation for the complete
eigenvalues-eigenvectors problem. In particular we show that the dependence of
all eigenvalues on $n$ is asymptotically linear for large $n$. A similar result
is obtained for more general (with specially scaled entries) left triangular
matrices as well. As an application we study ergodic properties of a family of
chaotic maps.
