Prove a covariance matrix is positive semidefinite

Question

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.

https://math.stackexchange.com/q/114072/321264 – StubbornAtom Jun 21 '20 at 16:14 — StubbornAtom, Jun 21 '20 at 16:14

score 8 · Accepted Answer · answered Feb 12 '16 at 11:57

8

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

answered Feb 12 '16 at 11:57

John B

16,854

score 1 · Answer 2 · answered Feb 12 '16 at 12:01

1

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.

answered Feb 12 '16 at 12:01

Ben Grossmann

225,327

Prove a covariance matrix is positive semidefinite

2 Answers2