Let $V := \mathbb{R}^n$, then the inner products on $V$ are in bijection with the set of symmetric positive definite matrices in $M_n(\mathbb{R})$. The bijection is given by sending a matrix $M$ to the inner product given by: $$\langle v,w\rangle_M := v^tMw\qquad\text{for all $v,w\in V$}$$ If $M,M'$ are two such matrices, then I will say that they give equivalent inner products on $V$ if there is an automorphism $T\in GL(V)$ such that $$\langle Tv,Tw\rangle_{M'} = \langle v,w\rangle_M$$ This translates into the condition: $$T^tMT = M$$ which is similar to asking for conjugacy classes of symmetric positive definite matrices, though not quite.
Is it possible to classify the equivalency classes of inner products on $V$?
References would be appreciated as well.
The classical result that any inner product admits an orthonormal basis exactly says that any two inner products are equivalent. In general, Sylvester's theorem says that over $\mathbb R$ symmetric bilinear forms are classified up to equivalence by rank and signature.