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I have difficulty especially proving that $f(\mathbf v, \mathbf v) \geq 0$ for all $\mathbf v$.

Thanks

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    $\begingroup$ The matrix $A$ isn't invertible? $\endgroup$
    – user63181
    Apr 16, 2014 at 14:43
  • $\begingroup$ For example, if $A=O$ is the zero-matrix then $f(u,v)$ is not an inner product, since $f(u,u)=0$ for all $u \in \mathbb{R}^n$. $\endgroup$
    – user35603
    Apr 16, 2014 at 14:54
  • $\begingroup$ yes you're right, A is invertible. I've edited the question $\endgroup$
    – rab2004
    Apr 16, 2014 at 15:21

1 Answer 1

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Let me tackle the bit that you find the most difficult. Let $x=A^Tv$. Note that

$$v^T A A^T v=x^T x = \sum_{i=1}^n x_i^2\geq 0.$$

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