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Given a random vector c with zero mean, the covariance matrix $\Sigma = E[cc^T]$. The following steps were given to prove that it is positive semidefinite.

$u^T\Sigma u = u^TE[cc^T]u = E[u^Tcc^Tu] = ||u^Tc|| \ge 0$

I don't understand how the expectation can equate to a norm.

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2 Answers 2

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Write instead $$ u^T\Sigma u = u^TE[cc^T]u = E[u^Tcc^Tu] = E[\|c^Tu\|^2] \ge 0. $$

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Note that for any vector $v,w$, the dot product is given by $v \cdot w = v^Tw$. Thus, we have $$ E[u^Tcc^Tu] = E[(u^Tc)(c^Tu)] = E[(u \cdot c)(c \cdot u)] = E[|u \cdot c|^2] \geq 0 $$ I'm not quite sure what they meant with the norm.

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