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I'm given that a quadratic form $q$ is positive definite if $q(v) > 0 \quad \forall 0 \neq v \in V$ and equivalently for bilinear form $\tau$. Does this mean that in a positive definite bilinear form there can exists $v,w \in V$ such that $\tau (v,w) \leq 0$ ?

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up vote 2 down vote accepted

Yes.

If $V=\mathbb{R},$ $\tau:(x,y)\mapsto xy$ is positive definite but $\tau(1,-1)=-1.$

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