Procesi and Formanek applied invariant theory (of \(GL(n)\), \(Sp(n)\), \(O(n)\)) to the study of p.i. algebras ( trace identities of \(n\times n\) matrices). The author adapts the theories developed by them to matrices with involution. Formanek reduced the problems of calculating the Poincaré series for the ring of generic \(n\times n\) matrices with trace \(\bar R\), or of its center \(\bar C\), to a problem in the theory of Schur functions; he showed how these series are related to those of the ring of \(n\times n\) generic matrices \(R\) (and to its center \(C\)). In this paper, the Poincaré series for the ring of generic \(n\times n\) matrices with trace and symplectic involution and the case of transpose involution are studied. These results are linked to rings of generic matrices with involution, without traces. These results help to demonstrate the importance of the (unsolved) combinatorial problem of expressing the symmetric functions \(S_ \lambda(x_ ix_ j^{-1}\mid i,j=1,\dots,n)\) as a linear combination of terms of the form \((x_ 1\dots x_ n)^{- \alpha} S_ \mu(x_ 1,\dots,x_ n)\). Procesi showed that if an algebra with trace satisfies the \(n\)-th Cayley-Hamilton equation, then it has a trace-preserving embedding into \(n\times n\) matrices. The author shows that if an algebra with trace and involution satisfies the same *-trace identities as \(n\times n\) matrices with (symplectic or transpose) involution, then it has a trace- and transpose-preserving embedding into \(n\times n\) matrices.