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?
Covariance matrix of a complex random variable
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statistics
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$C_{k\ell}=C_{k\ell}^{rr}+C_{k\ell}^{ii}+\mathrm i\cdot(C_{k\ell}^{ir}-C_{k\ell}^{ri})=C_{k\ell}^{rr}+C_{k\ell}^{ii}+\mathrm i\cdot(C_{k\ell}^{ir}-C_{\ell k}^{ir})$
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0Would this determine these degenerated cases? $\begin{array}{l}E(X_{1}Y_{1})=0\\ E(X_{2}Y_{2})=0\\E(X_{1}^{2})= E(X_{2}^{2})\\ E(Y_{1}^{2})= E(Y_{2}^{2})\\E(X_{1}X_{2})= E(Y_{1}Y_{2})\\E(X_{1}Y_{2})= E(Y_{1}X_{2})\end{array}$ – 2012-08-20