  General algebraic properties of the algebras of vector fields over quantum
linear groups $GL_q(N)$ and $SL_q(N)$ are studied. These quantum algebras
appears to be quite similar to the classical matrix algebra. In particular,
quantum analogues of the characteristic polynomial and characteristic identity
are obtained for them. The $q$-analogues of the Newton relations connecting two
different generating sets of central elements of these algebras (the
determinant-like and the trace-like ones) are derived. This allows one to
express the $q$-determinant of quantized vector fields in terms of their
$q$-traces.
