For a univariate analysis of trait k, y of Eq. (4) is a column vector of length N × 1 and W has dimension N × M. Assuming b is an M × 1 vector of random SNP effects for trait k, with distribution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{b}}\sim N\left( {0,{\bf{I}}_M\sigma _{b_k}^2} \right)$$\end{document}b~N0,IMσbk2, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{B}} = {\bf{I}}_M\sigma _{b_k}^2$$\end{document}B=IMσbk2 with IM is an identity matrix of dimension M. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{\epsilon }}$$\end{document}ϵ of Eq. (1) is a column vector of independent residual effects, with distribution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{\epsilon }}\sim N\left( {0,{\bf{I}}_N\sigma _{{\it{\epsilon }}_k}^2} \right)$$\end{document}ϵ~N0,INσϵk2, giving \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{R}} = {\bf{I}}_N\sigma _{\epsilon _k}^2$$\end{document}R=INσϵk2, with IN an identity matrix of dimension N. Substituting these expressions into Eq. (6) means that Eq. (7) can then be written as:8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{BLUP}}_k} = \left[ {{\bf{W}}_k\prime {\bf{W}}_k + {\bf{I}}_M\lambda _k} \right]^{ - 1}{\bf{W}}_k\prime {\bf{y}}_k$$\end{document}b^BLUPk=Wk′Wk+IMλk-1Wk′ykwith \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _k = \sigma _{\epsilon _k}^2{\mathrm{/}}\sigma _{{\it{b}}_k}^2$$\end{document}λk=σϵk2∕σbk2.