I am aware that certain special quantities of tensors (e.g. trace, det) are invariants (i.e. unchanging wrt coordinate system changes).
This wikipedia article says that the invariants of tensors are coefficients of the characteristic polynomial of the tensor. But are there not other ones? In particular, are any of the standard matrix norms invariants?
So it seems like one way to do this is to use contractions as tensor norms, which are known to be invariant.
For instance, one can use the double contraction for many of the tensors from the continuum mechanics as a norm: $$ ||D|| = \sqrt{D:D} = \sqrt{\text{tr}\left(DD^T\right)} $$ But, more generally (from here), one can use contractions. For instance, for a type (2,0)-tensor: $$ ||T||^2 = T_{ij} T^{ij}= g_{ik} g_{jl} T^{kl} T^{ij} $$ Or, for a type-(1,3) tensor (e.g. the Riemann curvature tensor), one could do: $$ ||R||^2 = R_{ijk}^{l} R_{mno}^p g^{im} g^{jn} g^{ko} g_{lp} = R_{ijk}^l R_l^{ijk} $$