  A classical linear group $G<GL(n)$ acts on $d$-tuples of $n\times n$ matrices
by simultaneous conjugation. Working over an infinite field of characteristic
different from two we establish that the ideal of free relations, i.e.
relations valid for matrices of any order, between generators for matrix O(n)-
and $\Sp(n)$-invariants is zero. We also prove similar result for invariants of
mixed representations of quivers.
  These results can be considered as a generalization of the characteristic
isomorphism ${\rm ch}:\Sym\to J$ between the graded ring
$\Sym=\otimes_{d=0}^{\infty} \Sym_d$, where $\Sym_d$ is the character group of
the symmetric group $S_d$, and the inverse limit $J$ with respect to $n$ of
rings of symmetric polynomials in $n$ variables.
  As a consequence, we complete the description of relations between generators
for O(n)-invariants as well as the description of relations for invariants of
mixed representations of quivers. We also obtain an independent proof of the
result that the ideal of free relations for $GL(n)$-invariants is zero, which
was proved by Donkin in [Math. Proc. Cambridge Philos. Soc. 113 (1993),
23--43].
