Let $M_{n\times n}(R)$ be the set of all $n \times n$ square matrices with real entries, and S the set of all symmetric invertible matrices in $M_{n\times n}(R)$.
For every $P$ in $S$, is there a basis $B$ of an n-dimensional inner product space $V$ with an inner product < , > such that $P$ is the matrix representation of the inner product with respect to the basis $B$?