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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 $$ \lim_{x_i \rightarrow \pm \infty} q(x_1,\ldots,x_n) = + \infty$$ for every $i$ while the remaining variables are fixed. Is it true that $q$ is necessarily positive definite?

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Take $x^2-1$, is it positive definite? – Raskolnikov May 14 '13 at 9:00
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it's not a quadratic form – mathaholic May 14 '13 at 9:07
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Silly me, you are of course right, but it doesn't matter, take $x^2-4xy+y^2$. – Raskolnikov May 14 '13 at 9:11

Consider the form $$q(x,y)=x^2+3y^2+4xy$$

In fact this is the form $$q(\bar x,\bar y)=\bar x^2-\bar y^2$$ in the basis $(1,0), (2,1)$.

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