Néron-Tate height pairing induces positive definite quadratic form

Question

Let $E$ be an elliptic curve.

Let $\langle \cdot , \cdot \rangle$ be the canonical height paring:

$\langle P , Q \rangle = 1/2(\hat{h}(P + Q) − \hat{h}(P ) − \hat{h}(Q))$

Let $P_1 , . . . P_n$ be a basis for $E(\Bbb Q)/{\rm Torsion}(E(\Bbb Q))$

I try to prove that the quadratic form $F=\sum_{1\le i,j\le n} \langle P_i , P_j \rangle x_ix_j$ is definite positive.

For $n=2$ it is easy to prove it using $\langle P , Q \rangle \le \sqrt[]{\hat{h}(P )\hat{h}(Q)}$, but how to generalize for any $n\ge 1$?

Matricial approach didn't help as it gets complicated to compute the determinant.

Thanks in advance for any hints of help.