Degrees of freedom - matrix diagonalization

Neitrino

Hi,
A symmetric 4x4 matrix has 10 independent components.
Let's say that matrix describes graviton h_mu_nu.
In general I can bring any symmetric matrix to diagonal form, so if can and I bring my h_mu_nu matrix to diagonal form where it has only 4 independent components ... than what happens to the rest 6 components...does this diagonalizing procedure (which in some sense means rotation in space-time and fixing my frame of reference ) "consumes" degrees of freedom ?

Could you please advise...

Thank you