# Is cov(x,y) and cov(y,x) the same thing?

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.

• Yes, covariance is "commutative". – Lord Shark the Unknown May 21 '18 at 6:09
• We don't need to. It's provided by definition. It just turns out this operation is commutative. – Alvin Lepik May 21 '18 at 6:10
• what do you mean by "... we need to write them in two different formats in the Cov matrix", what formats are you talking about? – MAN-MADE May 21 '18 at 6:14

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 $\rho_{\min\{i,\,j\}\max\{i,\,j\}}$.