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

Question

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.

Answer 1

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