On rational invariants of a set of symmetric matrices
A polynomial f is called an invariant of an operator A:\(L\to L\), with L a vector n-space, if for any regular linear operator D:\(L\to L\) one has \(f(A)=f(DAD^{-1})\). The ring of invariants of A is generated by the coefficients of the characteristic polynomial of A. The straightforward generalizations of this statement to the case of two operators (A,B) is not true. However, starting with some identities resulting from the Cayley-Hamilton theorem the author obtains the following (a) for \(k=1,2,...,n-1\) a condition is found for a pair (A,B) of symmetric operators acting in Euclidean n-space to have common invariant k-subspace (provided that A does not have multiple eigenvalues); (b) it is shown that the field of rational invariants of (A,B) is isomorphic to a subfield of a rational function field with \(n(n+3)/2\) generators consisting of elements symmetric with respect to the permutation group \(P_ n\); (c) it is shown that any rational invariant of \((g+2)\) symmetric operators \(A,B,C_ 1,C_ 2,...,C_ g\) can be expressed as a rational function of invariants of one or two operators that are taken for pairs (A,B), \((A,C_ 1)\), \((A,C_ 2),...,(A,C_ g)\), \((A,B+C_ 1)\), \((A,B+C_ 2),...,(A,B+C_ g)\).
scientific article; zbMATH DE number 3843933