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Let $$q(x_1,\ldots,x_n) = \sum_{i,j=1}^{n}a_{ij}{x_i}{x_j}, \quad a_{ij}\in \mathbb{R}.$$ be a quadratic form with real coefficients. Suppose that the limit is nonnegative in every direction. That is, for any unit vector $u$, $$ \lim_{t \rightarrow \infty} q(tu) \geq 0.$$ Is $q$ necessarily positive semidefinite? If the condition is strengthened to $$ \lim_{t \rightarrow \infty} q(tu) > 0,$$ is it necessary positive definite?

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up vote 1 down vote accepted

Hint: if the coefficients are constant, then you don't need the limit. Bilinearity allows to evacuate $t^2>0$, so all you get is

$\forall u\in\mathbb R^n\, q(u) = \langle Au,u\rangle\ge 0$ where $A$ is a square symetric matrix.

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