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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


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It is obviously true when $A$ is real. – akkkk Apr 16 '12 at 14:05
up vote 3 down vote accepted

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! – Y.Z Apr 16 '12 at 14:16

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