# About symmetric bilinear forms

I am confused about symmetric bilinear forms. Here is my question: Let b is a positive definite symmetric bilinear form on real vector space. Is b non-degenerate?

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I think $0$ is usually considered positive-definite in this context. –  tomasz Dec 23 '12 at 14:55
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## 1 Answer

Yes. To say that $b$ is non-degenerate is the same as to say that there is no $x$ such that the function $f_x: V \to V$ given by $$y \mapsto b(x, y)$$ is identically the zero function.

To say that $b$ is positive definite means that for any $x$, $b(x, x) > 0$. In particular, $f_x(x) = b(x, x) \neq 0$ and so $f_x$ is not identically the zero function.

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Isn't there any way to prove that? I am trying to show that is b symmetric bilinear respectively positive,negative and indefinite, b is non-degenerate and controversely while b is non-degenerate is b pos, neg or indefinite? –  Serkan Hafız Yaray Dec 23 '12 at 19:04
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