Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|cite|improve this question

1 Answer 1

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.