Complex covariance (${\mathrm{Cov}}_{\mathbb{C}}(Z,W)$)

For square-integrable complex random variables, the Hermitian covariance is ${\mathrm{Cov}}_{\mathbb{C}}(Z,W)=\mathbb{E}[(Z-\mathbb{E}Z)\overline{(W-\mathbb{E}W)}]$. Its diagonal is nonnegative and equals $\mathbb{E}|Z-\mathbb{E}Z{|}^2$. The conjugation distinguishes it from the pseudo-covariance.

For a centered complex random variable $Z=A+iB$, the pseudo-covariance is $P=\mathbb{E}{Z}^2$. Together with $V=\mathbb{E}|Z{|}^2$ it determines the real covariance matrix: $\mathbb{E}{A}^2=(V+\mathrm{Re}P)/2$, $\mathbb{E}{B}^2=(V-\mathrm{Re}P)/2$, and $\mathbb{E}AB=\mathrm{Im}P/2$.