Computation And Application Of Matrix Invariants

Symmetric polynomial is a basic example of the invariants.The main problems which invariant theory research are the set of generators of invariants,the defining relations of the generators of invariants,the structure of invariant ring,and so on.This paper give the set ofgenerators of one generic matrix of size nxn under the action of GL_n(C) by conjugation.i.e.improve that with the action of GL_n : g:x_ij→y_ij,i,j=1,…n.then the algebras of invariants k[x_ij]^GL_n is generated by coefficiernts of the characteristic polynomial of X .From this we can introduce that the algebras of invariants k[x_ij]^GL_n is generated by the traces tr(X),tr(X²),…,tr(Xⁿ)of the powers of X.