If k traits are measured on different individuals, with Nk observations for trait k, the elements of Eq. (4) become: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\bf y}^ \prime} = [{\bf y}\prime_{\bf 1}...{{\bf y}^ \prime_{k}}], {\bf{W}} = \left[ {\begin{array}{*{20}{c}} {{{\bf W}}_1} & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & {{\bf{W}}_k} \end{array}} \right]$$\end{document}y′=[y′1...yk′],W=W1000⋱000Wk, and R = diag[Rk] = diag\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ {{\bf{I}}_{N_k}\sigma _{\epsilon _k}^2} \right]$$\end{document}INkσϵk2, a diagonal matrix of length \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N = \mathop {\sum }\limits_k N_k$$\end{document}N= ∑kNk. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{B}} = {\bf{\Sigma }}_b \otimes {\bf{I}}_M$$\end{document}B=Σb⊗IM, where Σb is a k × k matrix, with diagonal elements \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{b_k}^2$$\end{document}σbk2 and off-diagonal elements the covariances of SNP effects between traits k and l, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{b_{k,l}}$$\end{document}σbk,l. For Kronecker products, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{B}}^{ - 1} = {\bf{\Sigma }}_b^{ - 1} \otimes {\bf{I}}_M$$\end{document}B-1=Σb-1⊗IM and substituting these expressions directly into Eq. (6) means that multi-trait BLUP solutions for b can be obtained in Eq. (7) as:9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{MT}} - {\mathrm{BLUP}}} = \left[ {{{{\bf W}{\prime}{\bf W}}} + {\bf{\Sigma }}_\epsilon {\bf{\Sigma }}_b^{ - 1} \otimes {\bf{I}}_M} \right]^{ - 1}{{{\bf W}{\prime}{\bf y}}}$$\end{document}b^MT-BLUP=W′W+ΣϵΣb-1⊗IM-1W′ywith \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{\Sigma }}_\epsilon = {\mathrm{diag}}\left[ {\sigma _{\epsilon _k}^2} \right]$$\end{document}Σϵ=diagσϵk2, a diagonal k × k matrix. For a two-trait example, Eq. (9) expands to:10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}{l}} {\widehat {\bf{b}}_{\mathrm{MT-BLUP}}} \hfill & = \hfill & {\left[ {\left[ {\begin{array}{*{20}{c}} {{\bf{W}}_1\prime {\bf{W}}_1} & 0 \\ 0 & {{\bf{W}}_2\prime {\bf{W}}_2} \end{array}} \right]} \right.} \hfill \\ {} \hfill & + \hfill & {\left. {\left[ {\begin{array}{*{20}{c}} {{\bf{I}}_M\sigma _{\epsilon _1}^2} & 0 \\ 0 & {{\bf{I}}_M\sigma _{\epsilon _2}^2} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\bf{I}}_M\sigma _{b_1}^2} & {{\bf{I}}_M\sigma _{b_{1,2}}} \\ {{\bf{I}}_M\sigma _{b_{2,1}}} & {{\bf{I}}_M\sigma _{b_2}^2} \end{array}} \right]^{ - 1}} \right]^{ - 1}} \hfill \\ {} \hfill & {} \hfill & {\left[ {\begin{array}{*{20}{c}} {{\bf{W}}_1} & 0 \\ 0 & {{\bf{W}}_2} \end{array}} \right]\prime \left[ {\begin{array}{*{20}{c}} {{\bf{y}}_1} \\ {{\bf{y}}_2} \end{array}} \right]} \hfill \end{array}$$\end{document}b^MT-BLUP=W1′W100W2′W2+IMσϵ1200IMσϵ22IMσb12IMσb1,2IMσb2,1IMσb22-1-1W100W2′y1y2