Second, we consider the off-diagonal elements of VSBLUP, which are comprised of the covariance of BLUP SNP solutions among the k traits. These can again be approximated from theory given the covariance of genetic effects among traits k and l is cov(bk, bl) = rGhkhl/M, with rG the genetic correlation, and given the squared correlation between the true genetic effects of the SNPs, bk, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}$$\end{document}b^BLUPk which is given by Eq. (17) as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{{\bf{b}}_k,\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}}^2 = \frac{{R_k^2}}{M}{\mathrm{/}}\frac{{h_k^2}}{M} = R_k^2{\mathrm{/}}h_k^2$$\end{document}Rbk,b^BLUPk2=Rk2M∕hk2M=Rk2∕hk2. The covariance of BLUP SNP predictors is then:19\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {\widehat {\bf{b}}_{{\rm BLUP}_{k{\prime}}},\widehat {\bf{b}}_{{\rm BLUP}_l}} \right) = \frac{{R_k^2}}{{h_k^2}} \cdot \frac{{R_l^2}}{{h_l^2}}{\mathrm{cov}}\left( {{\bf{b}}_k,{\bf{b}}_{\mathit{l}}} \right) = \frac{{r_{\mathrm{G}}R_k^2R_l^2}}{{h_kh_lM}}$$\end{document}covb^BLUPk′,b^BLUPl=Rk2hk2⋅Rl2hl2covbk,bl=rGRk2Rl2hkhlM