For an arbitrary $k$-linear form $P$ on $R^n$ of the form: $P(x_1,..,x_k)=\sum_{i_1,...,i_k=1}^n a_{i_1,...,i_k}x_{1i_1} ...x_{ki_k}$ we put $I(P):=\sum_{i_1,...,i_k=1}^n a_{i_1,...,i_k}^2.$ Is this quantity invariant with respect to ortogonal group? In other words, let $P$ be $k$ linear form, $f\in O(n)$ and $P_f(x_1,...,x_k):=P(f(x_1),...,f(x_k))$. Is it then $I(P)=I(P_f) ?$
I know that it is true for $k=2$. In that case $P(x_1,x_2)=x_1^TAx_2$, P_f(x_1',x_2')= x_1'^TC^TACx_2' (in matrix denotation). Then $Tr(A A^T)=\sum_{i_1,i_2}^n a_{i_1i_2}^2$ and $Tr((C^TAC)(C^TAC)^T)=Tr(C^{-1}AA^TC)=Tr(AA^T)=\sum_{i_1,i_2}^n a_{i_1i_2}^2$ ( since $C^{-1}=C^T$).
Thanks.