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There are two $p\times 1$ random vectors. I was wondering if the following statement holds. If the expectation of the outer product is a $p\times p$ zero matrix, the expectation of its inner product is also zero(scalar).

$E(XY^T)=0\Rightarrow E(X^T Y)=0$

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    Note that $X^TY=\mathrm{tr}(XY^T)$. – 2017-02-17
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    Wow, thank you so much! – 2017-02-17
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    You actually have the stronger result, $E(X^TY)=0$ implies either $E(X)=0$ or $E(Y)=0$. – 2017-02-17
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    @ThomasAndrews This is not true. Let $p$=1, $X=X_1\sim B(1/2)$, $Y=Y_1=1-X$, then $E(X^TY)=E(XY)=0$ without $EX=0$ and $EY=0$. – 2017-02-17
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    Whops, yep, was assuming independence. @NCh – 2017-02-17

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