If $V$ is a vector space with finite dimension and for some symmetric bilinear form $f: V \times V \rightarrow \mathbb{R}$, how to I show that $f$ defines an inner product iff the unique real canonical form of $f$ is $I_n$.
So I know that we can represent the bilinear form as a quadratic form and from sylvesters law of inertia we have that there must be exactly one canonical (Thus proving uniqueness). However, for this how do I start showing that the basis elements are all orthogonal with full rank?
Any help would be greatly appreciated!
Best Answer
Take any basis $\{{e_i\}}$ of $V$; there should be no problem in constructing one. This enables one to construct the symmetric matrix: $f_{i j}:=f(e_i,e_j)$. Diagonalise $f_{i j}$ using the eigenvalues. If all the eigenvalues are positive, it is an inner product.