# Why covariance matrix is positive semi definite?

30/08/2011

Today it is a simple question for me. But yesterday, I was bothered by it.

**Problem**: Let x be a random vector with mean as and

Now prove is positive semi definite.

**Proof**: Let be an arbitrary vector (not random vector). Then

Q.E.D.

**Highlight**:

- The matrix is p.s.d and of rank-1. But we can’t simply say is p.s.d, and of course is not of rank-1.
- Need not to use the definition of expectation to prove, but need use the definition of positive definite matrices.
- For scalar cases, it is easy to see because

**Further question**: when does ?? when the elements of are independent? linear or statistically independent? So if x is independent, we have the covariance positive definite instead of positive semi-definite?

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