Problem. I have a real symmetric $n \times n$ matrix $A$ and would like to compute a set of real numbers
$f(A) = (x_1, \ldots, x_m) \in \mathbb R^m$
which are invariant under multiplication of $A$ with an orthogonal matrix, that is
$f(QA) = f(A) \quad\iff\quad Q^TQ = I$.
It is important that the invariants are invariant to orthogonal transformations only. I.e., they should change when $Q$ is not orthogonal.
Comments. Wikipedia says that "... the singular values of the bilinear forms, represented as a matrix, form a complete set of invariants for bilinear forms under orthogonal substitutions." (https://en.wikipedia.org/wiki/Singular_value_decomposition)
Are the singular values what I am looking for? Is "orthogonal substitution" multiplying $A$ with an orthogonal $Q$?
It follows from $(QA)^TQA = A^TA$ that $QA$ and $A$ have the same singular values, so they seem a good candidate indeed. But are they invariant under other more general (than orthogonal) transformations? Could someone point me to some references about this topic?