  We establish the eigenvalue interlacing property (i.e. the smallest real
eigenvalue of a matrix is less than the smallest real eigenvalue of any its
principal submatrix) for the class of matrices, introduced by Kotelyansky (all
principal and all almost principal minors of these matrices are positive). We
show that certain generalizations of Kotelyansky and totally positive matrices
also possess this property. We prove some interlacing inequalities for the
other eigenvalues of Kotelyansky matrices.
