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We say an Hermitian matrix $A$ is positive if $ \bar{z}^tAz=\sum_{i,j=1}^na_{ij}\bar{z}_iz_j>0,\quad \forall z\neq 0.$

But if we have $z^tA\bar{z}=\sum_{i,j=1}^na_{ij}z_i\bar{z}_j>0,\quad \forall z\neq 0.$ can we say that $A$ is positive? Prove or counterexample

Thanks!

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    It is obviously true when $A$ is real.2012-04-16

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Replace $z$ in the second line by $w$ and then choose $w=\bar z$ to see that the two statements are equivalent.

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    Thanks! And I can't believe that I asked such a silly question!2012-04-16