Covariance matrix of a complex random variable

Question

If one considers a complex random variable as the joint random variable of the real and complex part, the covariance matrix of two complex random variables $Z_{1}=X_{1}+iY_{1}$ and $Z_{2}=X_{2}+iY_{2}$ becomes a $4\times 4$ matrix $$\begin{pmatrix}C^{(rr)}&C^{(ri)}\\C^{(ir)}&C^{(ii)}\end{pmatrix}$$ $$C^{(rr)}_{ij}=E((X_{i}-E(X_{i}))(X_{j}-E(X_{j})))$$ $$C^{(ri)}_{ij}=E((X_{i}-E(X_{i}))(Y_{j}-E(Y_{j})))$$ $$C^{(ir)}_{ij}=E((Y_{i}-E(Y_{i}))(X_{j}-E(X_{j})))$$ $$C^{(ii)}_{ij}=E((Y_{i}-E(Y_{i}))(Y_{j}-E(Y_{j})))$$ However the covariance matrix of two complex random variables is often defined as a $2\times 2$ matrix $$C_{ij}=E((Z_{i}-E(Z_{i}))(Z_{j}-E(Z_{j}))^{\ast})$$ How are these two concepts related?

Answer 1

$$C_{k\ell}=C_{k\ell}^{rr}+C_{k\ell}^{ii}+\mathrm i\cdot(C_{k\ell}^{ir}-C_{k\ell}^{ri})=C_{k\ell}^{rr}+C_{k\ell}^{ii}+\mathrm i\cdot(C_{k\ell}^{ir}-C_{\ell k}^{ir})$$