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I am learning the concepts of covariance and covariance matrix. It seems to me that:

Cov(x, y) = E((x - E(x))(y-E(y))) = E((y-E(y))(x-E(x))) = Cov(y,x)

Is that the case? If so, why do we need to write them in two different formats in the Cov matrix.

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    $\begingroup$ Yes, covariance is "commutative". $\endgroup$ – Lord Shark the Unknown May 21 '18 at 6:09
  • $\begingroup$ We don't need to. It's provided by definition. It just turns out this operation is commutative. $\endgroup$ – Alvin Lepik May 21 '18 at 6:10
  • $\begingroup$ what do you mean by "... we need to write them in two different formats in the Cov matrix", what formats are you talking about? $\endgroup$ – MAN-MADE May 21 '18 at 6:14
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The covariance matrix of multiple variables is indeed symmetric, but we still need to fill in the matrix. When we write facts about it, it's more convenient to write $\rho_{ij}$ than $\rho_{\min\{i,\,j\}\max\{i,\,j\}}$.

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