Appendix
We need the following well-known result by Penrose (1955) on solvability and general solution of linear matrix equation.
Lemma 1
The linear matrix equation \(\mathbf{Z}\mathbf{A}= \mathbf{B}\) is consistent if and only if \(r[\mathbf{A}^{\prime }, \, \mathbf{B}^{\prime }\,] = r(\mathbf{A}^{\prime }),\) or equivalently, \(\mathbf{B}\mathbf{A}^{+}\mathbf{A}= \mathbf{B}.\) In this case, the general solution of \(\mathbf{Z}\mathbf{A}= \mathbf{B}\) can be written in the following parametric form \(\mathbf{Z}= \mathbf{B}\mathbf{A}^{+} + \mathbf{U}\mathbf{A}^{\!\perp },\) where \(\mathbf{U}\) is an arbitrary matrix.
We also need the following formulas on ranks of matrices; see, e.g., Marsaglia and Styan (1974), Tian (2004).
Lemma 2
Let \(\mathbf{A}\in {\mathbb {R}}^{m \times n},\)
\( \mathbf{B}\in {\mathbb {R}}^{m \times k},\) and \(\mathbf{C}\in {\mathbb {R}}^{l \times n}.\) Then
$$\begin{aligned} r[\, \mathbf{A}, \, \mathbf{B}\, ]&= r(\mathbf{A})+ r(\mathbf{E}_{\mathbf{A}}\mathbf{B})= r(\mathbf{B}) + r(\mathbf{E}_{\mathbf{B}}\mathbf{A}), \end{aligned}$$
(56)
$$\begin{aligned} r\!\left[ \begin{array}{c} \mathbf{A}\\ \mathbf{C}\end{array}\right]&= r(\mathbf{A}) + r(\mathbf{C}\mathbf{F}_{\mathbf{A}}) = r(\mathbf{C}) + r(\mathbf{A}\mathbf{F}_{\mathbf{C}}), \end{aligned}$$
(57)
$$\begin{aligned} r\!\left[ \begin{array}{c@{\quad }c} \mathbf{A}\mathbf{A}^{\prime } \ &{} \mathbf{B}\\ \mathbf{B}^{\prime } &{} \mathbf{0}\end{array} \right]&= r[\, \mathbf{A}, \, \mathbf{B}\,] + r(\mathbf{B}). \end{aligned}$$
(58)
If \({\fancyscript{R}}(\mathbf{A}_1^{\prime }) \subseteq {\fancyscript{R}}(\mathbf{B}_1^{\prime })\), \({\fancyscript{R}}(\mathbf{A}_2) \subseteq {\fancyscript{R}}(\mathbf{B}_1)\), \({\fancyscript{R}}(\mathbf{A}_2^{\prime }) \subseteq {\fancyscript{R}}(\mathbf{B}_2^{\prime })\) and \({\fancyscript{R}}(\mathbf{A}_3) \subseteq {\fancyscript{R}}(\mathbf{B}_2),\) then
$$\begin{aligned} r(\mathbf{A}_1\mathbf{B}_1^+\mathbf{A}_2)&= r\!\left[ \begin{array}{c@{\quad }c} \mathbf{B}_1 &{} \mathbf{A}_2 \\ \mathbf{A}_1 &{} \mathbf{0}\end{array} \right] - r(\mathbf{B}_1), \end{aligned}$$
(59)
$$\begin{aligned} r(\mathbf{A}_1\mathbf{B}_1^+\mathbf{A}_2\mathbf{B}_2^+\mathbf{A}_3)&= r\!\left[ \begin{array}{c@{\quad }c@{\quad }c} \mathbf{0}&{} \mathbf{B}_2 &{} \mathbf{A}_3 \\ \mathbf{B}_1 &{} \mathbf{A}_2 &{} \mathbf{0}\\ \mathbf{A}_1 &{} \mathbf{0}&{} \mathbf{0}\end{array} \right] - r(\mathbf{B}_1) - r(\mathbf{B}_2). \end{aligned}$$
(60)
Lemma 3
Let
$$\begin{aligned} f(\mathbf{Z}) = \left( \, \mathbf{Z}\mathbf{C}+ \mathbf{D}\,\right) \!\mathbf{M}\!\left( \,\mathbf{Z}\mathbf{C}+ \mathbf{D}\right) ^{\prime } - \mathbf{N}\ \ s.t. \ \ \mathbf{Z}\mathbf{A}= \mathbf{B}, \end{aligned}$$
(61)
where \(\mathbf{A}\in {{\mathbb {R}}}^{p \times q}\), \(\mathbf{B}\in {{\mathbb {R}}}^{n \times q}\), \(\mathbf{C}\in {{\mathbb {R}}}^{p\times m},\)
\(\mathbf{D}\in {{\mathbb {R}}}^{n\times m},\) and \(\mathbf{N}= \mathbf{N}^{\prime } \in {{\mathbb {R}}}^{n\times n}\) are given, \(\mathbf{M}\in {{\mathbb {R}}}^{m \times m}\) is nnd, and the matrix equation \(\mathbf{Z}\mathbf{A}= \mathbf{B}\) is consistent. Then, there always exists a solution \(\mathbf{Z}_0\) of \(\mathbf{Z}_0\mathbf{A}= \mathbf{B}\) such that
$$\begin{aligned} f(\mathbf{Z}) \succcurlyeq f(\mathbf{Z}_0) \end{aligned}$$
(62)
holds for all solutions of \(\mathbf{Z}\mathbf{A}= \mathbf{B}\). In this case, the matrix \(\mathbf{Z}_0\) satisfying (62) is determined by the following consistent matrix equation
$$\begin{aligned} \mathbf{Z}_0[\,\mathbf{A}, \, \mathbf{C}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp } \,] = [\,\mathbf{B}, \, -\mathbf{D}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp } \,]. \end{aligned}$$
(63)
The general expression of \(\mathbf{Z}_0\) and the corresponding \(f(\mathbf{Z}_0)\) are given by
$$\begin{aligned} \mathbf{Z}_0&= \mathop {\mathrm{argmin}}\limits _{\mathbf{Z}\mathbf{A}= \mathbf{B}}\! f(\mathbf{Z}) = [\,\mathbf{B}, \, -\mathbf{D}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp } \,][\,\mathbf{A}, \, \mathbf{C}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp } \,]^+ + \mathbf{T}[\,\mathbf{A}, \, \mathbf{C}\mathbf{M}\mathbf{C}^{\prime }\,]^{\!\perp },\nonumber \\ \end{aligned}$$
(64)
$$\begin{aligned} f(\mathbf{Z}_0)&= \min _{\!\!\!\!\!\mathbf{Z}\mathbf{A}= \mathbf{B}}\! f(\mathbf{Z}) = \mathbf{S}\mathbf{M}\mathbf{S}^{\prime } - \mathbf{S}\mathbf{M}\mathbf{C}^{\prime }(\mathbf{A}^{\!\perp }\mathbf{C}\mathbf{M}\mathbf{C}^{\prime } \mathbf{A}^{\!\perp })^{+}\mathbf{C}\mathbf{M}\mathbf{S}^{\prime } - \mathbf{N}, \end{aligned}$$
(65)
and
$$\begin{aligned} f(\mathbf{Z}) - f(\mathbf{Z}_0)&= (\mathbf{Z}\mathbf{C}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp } + \mathbf{D}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp })(\mathbf{A}^{\!\perp } \mathbf{C}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp })^{+}\nonumber \\&\times (\mathbf{Z}\mathbf{C}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp } + \mathbf{D}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp })^{\prime }, \end{aligned}$$
(66)
where \(\mathbf{S}= \mathbf{B}\mathbf{A}^{+}\mathbf{C}+ \mathbf{D},\) and \(\mathbf{T}\in {{\mathbb {R}}}^{n\times p}\) is arbitrary.
Proof of Lemma 3
Substituting the general solution \(\mathbf{Z}= \mathbf{B}\mathbf{A}^{+} + \mathbf{U}\mathbf{A}^{\!\perp }\) of \(\mathbf{Z}\mathbf{A}= \mathbf{B}\) into \(f(\mathbf{Z})\) in (61), we obtain
$$\begin{aligned} f(\mathbf{Z})&= \left( \, \mathbf{U}\mathbf{A}^{\!\perp }\mathbf{C}+ \mathbf{S}\,\right) \!\mathbf{M}\!\left( \, \mathbf{U}\mathbf{A}^{\!\perp }\mathbf{C}+ \mathbf{S}\right) ^{\prime } - \mathbf{N}. \end{aligned}$$
(67)
Under \(\mathbf{M}\succcurlyeq \mathbf{0}\), the following two results
$$\begin{aligned} \mathbf{A}^{\!\perp }\mathbf{C}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp } \succcurlyeq \mathbf{0}\quad \mathrm{and} \quad {\fancyscript{R}}(\mathbf{A}^{\!\perp }\mathbf{C}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp }) = {\fancyscript{R}}(\mathbf{A}^{\!\perp }\mathbf{C}\mathbf{M}\mathbf{C}^{\prime }) = {\fancyscript{R}}(\mathbf{A}^{\!\perp }\mathbf{C}\mathbf{M}) \end{aligned}$$
(68)
hold. Hence, \((\mathbf{A}^{\!\perp }\mathbf{C}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp }) (\mathbf{A}^{\!\perp }\mathbf{C}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp })^{+} \mathbf{A}^{\!\perp }\mathbf{C}\mathbf{M}\mathbf{S}^{\prime } = \mathbf{A}^{\!\perp }\mathbf{C}\mathbf{M}\mathbf{S}^{\prime }\) holds as well. In this case, it is easy to verify that (67) can be decomposed as
$$\begin{aligned}&f(\mathbf{Z}) \nonumber \\&\quad = (\mathbf{U}\mathbf{A}^{\!\perp }\mathbf{C}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp } \! + \mathbf{S}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp })(\mathbf{A}^{\!\perp } \mathbf{C}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp })^{+} (\mathbf{U}\mathbf{A}^{\!\perp }\mathbf{C}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp } \! + \mathbf{S}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp })^{\prime } \nonumber \\&\qquad + \mathbf{S}\mathbf{M}\mathbf{S}^{\prime } - \mathbf{S}\mathbf{M}\mathbf{C}^{\prime }(\mathbf{A}^{\!\perp }\mathbf{C}\mathbf{M}\mathbf{C}^{\prime } \mathbf{A}^{\!\perp })^{+}\mathbf{C}\mathbf{M}\mathbf{S}^{\prime } - \mathbf{N}. \end{aligned}$$
(69)
Substituting \(\mathbf{U}\mathbf{A}^{\!\perp } = \mathbf{Z}- \mathbf{B}\mathbf{A}^{+}\) into (69) and simplifying yields
$$\begin{aligned} f(\mathbf{Z})&= (\mathbf{Z}\mathbf{C}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp } + \mathbf{D}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp })(\mathbf{A}^{\!\perp } \mathbf{C}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp })^{+}(\mathbf{Z}\mathbf{C}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp } + \mathbf{D}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp })^{\prime } \nonumber \\&\quad + \ \mathbf{S}\mathbf{M}\mathbf{S}^{\prime } - \mathbf{S}\mathbf{M}\mathbf{C}^{\prime } (\mathbf{A}^{\!\perp }\mathbf{C}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp })^+\mathbf{C}\mathbf{M}\mathbf{S}^{\prime } - \mathbf{N}. \end{aligned}$$
(70)
Note that
$$\begin{aligned} (\mathbf{Z}\mathbf{C}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp } + \mathbf{D}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp })(\mathbf{A}^{\!\perp }\mathbf{C}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp })^{+} (\mathbf{Z}\mathbf{C}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp } + \mathbf{D}\mathbf{M}\mathbf{C}^{\prime } \mathbf{A}^{\!\perp })^{\prime } \succcurlyeq \mathbf{0}\end{aligned}$$
in (70). Hence, \(f(\mathbf{Z}) \succcurlyeq \mathbf{S}\mathbf{M}\mathbf{S}^{\prime } - \mathbf{S}\mathbf{M}\mathbf{C}^{\prime }(\mathbf{A}^{\!\perp }\mathbf{C}\mathbf{M}\mathbf{C}^{\prime } \mathbf{A}^{\!\perp })^+\mathbf{C}\mathbf{M}\mathbf{S}^{\prime } - \mathbf{N}\) holds for all \(\mathbf{Z}\). Under (68), it is obvious that
$$\begin{aligned}&(\mathbf{Z}\mathbf{C}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp } + \mathbf{D}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp })(\mathbf{A}^{\!\perp } \mathbf{C}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp })^{+} (\mathbf{Z}\mathbf{C}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp } + \mathbf{D}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp })^{\prime } = \mathbf{0}\nonumber \\&\quad \Leftrightarrow \mathbf{Z}\mathbf{C}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp } + \mathbf{D}\mathbf{M}\mathbf{C}^{\prime }\mathbf{A}^{\!\perp }= \mathbf{0} \end{aligned}$$
(71)
holds. Since \({\fancyscript{R}}(\mathbf{A}^{\!\perp }\mathbf{C}\mathbf{M}\mathbf{D}^{\prime }) \subseteq {\fancyscript{R}}(\mathbf{A}^{\!\perp }\mathbf{C}\mathbf{M}\mathbf{C}^{\prime })\), the linear matrix equation in (71) is always consistent. Combining it with the given consistent matrix equation \(\mathbf{Z}\mathbf{A}= \mathbf{B}\) yields (63), which is consistent again. Solving the equation yields (64). Eqs. (65) and (66) follow from (70) and (71). \(\square \)
Proof of Theorem 1
It is obvious that
$$\begin{aligned} E(\mathbf{L}\mathbf{y}) = E({\pmb {\phi }}) \Leftrightarrow \mathbf{L}\widehat{\mathbf{X}} {\pmb {\alpha }}= \mathbf{F}{\pmb {\alpha }}\quad \text{ for } \text{ all } {\pmb {\alpha }} \Leftrightarrow \mathbf{L}\widehat{\mathbf{X}} = \mathbf{F}. \end{aligned}$$
From Lemma 1, the matrix equation \(\mathbf{L}\widehat{\mathbf{X}} = \mathbf{F}\) is solvable for \(\mathbf{L}\) if and only if (10) holds. Also, note that the corresponding \(\mathbf{L}\mathbf{y}- {\pmb {\phi }}\) can be written as \(\mathbf{L}\mathbf{y}- {\pmb {\phi }}= (\mathbf{L}\widehat{\mathbf{X}} - \mathbf{F}){\pmb {\alpha }}+ ( \mathbf{L}\mathbf{X}- \mathbf{G}){\pmb {\gamma }}+ (\mathbf{L}- \mathbf{H}){\pmb {\varepsilon }}\). Hence
$$\begin{aligned} Cov(\mathbf{L}\mathbf{y}- {\pmb {\phi }})&= Cov[( \mathbf{L}\mathbf{X}- \mathbf{G}){\pmb {\gamma }}+ (\mathbf{L}- \mathbf{H}){\pmb {\varepsilon }}] \nonumber \\&= [\, \mathbf{L}\mathbf{X}- \mathbf{G}, \, \mathbf{L}- \mathbf{H}\,] {\pmb {\Sigma }}[\, \mathbf{L}\mathbf{X}- \mathbf{G}, \, \mathbf{L}- \mathbf{H}\,]^{\prime } \nonumber \\&= ( \mathbf{L}\widetilde{\mathbf{X}} - \mathbf{J}){\pmb {\Sigma }}(\mathbf{L}\widetilde{\mathbf{X}} - \mathbf{J})^{\prime } := f(\mathbf{L}). \end{aligned}$$
(72)
Thus, (11) is equivalent to finding a solution \(\mathbf{L}_0\) of \(\mathbf{L}_0\widehat{\mathbf{X}} = \mathbf{F}\) such that
$$\begin{aligned} f(\mathbf{L}) \succcurlyeq f(\mathbf{L}_0) \ \ \text{ s.t. } \ \ \mathbf{L}\widehat{\mathbf{X}} = \mathbf{F} \end{aligned}$$
(73)
holds in the Löwner partial ordering. Since \({\pmb {\Sigma }}\succcurlyeq \mathbf{0}\), (72) is special case of (61). From Lemma 3, there always exists a solution \(\mathbf{L}_0\) of \(\mathbf{L}_0\widehat{\mathbf{X}} = \mathbf{F}\) such that \(f(\mathbf{L}) \succcurlyeq f(\mathbf{L}_0)\) holds for all solutions of \(\mathbf{L}\widehat{\mathbf{X}} = \mathbf{F}\), and the \(\mathbf{L}_0\) is determined by the matrix equation \(\mathbf{L}_0[\,\widehat{\mathbf{X}}, \, \widetilde{\mathbf{X}}{\pmb {\Sigma }}\widetilde{\mathbf{X}}^{\prime }\widehat{\mathbf{X}}^{\!\perp } \,] = [\,\mathbf{F}, \, \mathbf{J}{\pmb {\Sigma }}\widetilde{\mathbf{X}}^{\prime }\widehat{\mathbf{X}}^{\!\perp } \,]\), or equivalently,
$$\begin{aligned} \mathbf{L}_0[\,\widehat{\mathbf{X}}, \, \mathbf{V}\widehat{\mathbf{X}}^{\!\perp } \,] = [\,\mathbf{F}, \, (\mathbf{G}\mathbf{V}_1 + \mathbf{H}\mathbf{V}_2)\widehat{\mathbf{X}}^{\!\perp } \,], \end{aligned}$$
(74)
establishing the matrix equation in (11), where \(\widetilde{\mathbf{X}}{\pmb {\Sigma }}\widetilde{\mathbf{X}}^{\prime } = \mathbf{V}\) and \(\mathbf{J}{\pmb {\Sigma }}\widetilde{\mathbf{X}}^{\prime } = \mathbf{G}\mathbf{V}_1 + \mathbf{H}\mathbf{V}_2\). This equation is always consistent, and the general solution of the equation is given in (12).
Taking covariance operation of (12) yields (13). Also from (6) and (12), the covariance matrix between \(\mathrm{BLUP}({\pmb {\phi }})\) and \({\pmb {\phi }}\) is
$$\begin{aligned}&Cov\{\,\mathrm{BLUP}({\pmb {\phi }}), \, {\pmb {\phi }}\,\}\\&\quad = Cov\!\left\{ ([\,\mathbf{F}, \, (\mathbf{G}\mathbf{V}_1 + \mathbf{H}\mathbf{V}_2)\widehat{\mathbf{X}}^{\!\perp } \,]\mathbf{W}^{+} + \mathbf{U}\mathbf{W}^{\!\perp })\mathbf{y}, \,\mathbf{G}{\pmb {\gamma }}+ \mathbf{H}{\pmb {\varepsilon }}\right\} \\&\quad = ([\,\mathbf{F}, \, (\mathbf{G}\mathbf{V}_1 + \mathbf{H}\mathbf{V}_2)\widehat{\mathbf{X}}^{\!\perp } \,]\mathbf{W}^{+} + \mathbf{U}\mathbf{W}^{\!\perp }) Cov\left( [\, \mathbf{X}, \,\mathbf{I}_n]\left[ \! \begin{array}{c} {\pmb {\gamma }}\\ {\pmb {\varepsilon }}\end{array} \!\right] , \, \left[ \! \begin{array}{c} {\pmb {\gamma }}\\ {\pmb {\varepsilon }}\end{array} \!\right] \right) [\,\mathbf{G}, \ \mathbf{H}\,]^{\prime }\\&\quad = [\,\mathbf{F}, \, (\mathbf{G}\mathbf{V}_1 + \mathbf{H}\mathbf{V}_2)\widehat{\mathbf{X}}^{\!\perp } \,]\mathbf{W}^+(\mathbf{G}\mathbf{V}_1 + \mathbf{H}\mathbf{V}_2)^{\prime } \ \ \ \text{(by } \text{(1)--(4)) }, \end{aligned}$$
establishing (14). Further by (65),
$$\begin{aligned}&Cov({\pmb {\phi }}- \mathbf{L}_0\mathbf{y})\\&\quad = (\mathbf{F}\widehat{\mathbf{X}}^{+}\widetilde{\mathbf{X}} - \mathbf{J})\left( {\pmb {\Sigma }}-{\pmb {\Sigma }}\widetilde{\mathbf{X}}^{\prime } (\widehat{\mathbf{X}}^{\!\perp }\widetilde{\mathbf{X}}{\pmb {\Sigma }}\widetilde{\mathbf{X}}^{\prime }\widehat{\mathbf{X}}^{\!\perp })^{+} \widetilde{\mathbf{X}}{\pmb {\Sigma }}\right) (\mathbf{F}\widehat{\mathbf{X}}^{+}\widetilde{\mathbf{X}} - \mathbf{J})^{\prime }\\&\quad = (\mathbf{F}\widehat{\mathbf{X}}^{+}\widetilde{\mathbf{X}} - \mathbf{J}) \left( {\pmb {\Sigma }}-\left[ \!\begin{array}{c}\mathbf{V}_1\\ \mathbf{V}_2 \end{array}\!\right] (\widehat{\mathbf{X}}^{\!\perp }\mathbf{V}\widehat{\mathbf{X}}^{\!\perp })^{+} [\,\mathbf{V}^{\prime }_1, \, \mathbf{V}^{\prime }_2 \,] \right) (\mathbf{F}\widehat{\mathbf{X}}^{+}\widetilde{\mathbf{X}} - \mathbf{J})^{\prime }, \end{aligned}$$
establishing (16).\(\square \)
Proof of Corollary 1
Note that from (12) that
$$\begin{aligned}{}[\mathbf{F}, \, (\mathbf{G}\mathbf{V}_1 + \mathbf{H}\mathbf{V}_2)\widehat{\mathbf{X}}^{\!\perp } \,]\mathbf{W}^{+} = [\,\mathbf{F}, \, \mathbf{0}\,]\mathbf{W}^+ + [\,\mathbf{0}, \, \mathbf{G}\mathbf{V}_1\widehat{\mathbf{X}}^{\!\perp } \,]\mathbf{W}^+ + [\mathbf{0}, \, \mathbf{H}\mathbf{V}_2\widehat{\mathbf{X}}^{\!\perp } \,]\mathbf{W}^{+}. \end{aligned}$$
Hence (12) can be rewritten as the sum
$$\begin{aligned} \mathrm{BLUP}({\pmb {\phi }})&= \left( [\,\mathbf{F}, \, \mathbf{0}\,]\mathbf{W}^+ + \mathbf{U}_1\mathbf{W}^{\!\perp } \right) \mathbf{y}+ \left( [\,\mathbf{0}, \, \mathbf{G}\mathbf{V}_1\widehat{\mathbf{X}}^{\!\perp } \,]\mathbf{W}^+ + \mathbf{U}_2\mathbf{W}^{\!\perp } \right) \mathbf{y}\\&+ \left( [\,\mathbf{0}, \, \mathbf{H}\mathbf{V}_2\widehat{\mathbf{X}}^{\!\perp } \,]\mathbf{W}^+ + \mathbf{U}_3\mathbf{W}^{\!\perp } \right) \mathbf{y}\\&= \mathrm{BLUE}(\mathbf{F}{\pmb {\alpha }}) + \mathrm{BLUP}(\mathbf{G}{\pmb {\gamma }}) + \mathrm{BLUP}(\mathbf{H}{\pmb {\varepsilon }}), \end{aligned}$$
establishing (17). We obtain from (12) that the covariance matrix between \(\mathrm{BLUE}(\mathbf{F}{\pmb {\alpha }})\) and \( \mathrm{BLUP}(\mathbf{G}{\pmb {\gamma }}+ \mathbf{H}{\pmb {\varepsilon }})\) is
$$\begin{aligned}&Cov\{\, \mathrm{BLUE}(\mathbf{F}{\pmb {\alpha }}), \, \mathrm{BLUP}(\mathbf{G}{\pmb {\gamma }}+ \mathbf{H}{\pmb {\varepsilon }}) \,\} \nonumber \\&\quad = Cov\left\{ \, ([\,\mathbf{F}, \, \mathbf{0}\,]\mathbf{W}^+ + \mathbf{U}_1\mathbf{W}^{\!\perp })\mathbf{y}, \ ([\,\mathbf{0}, \, (\mathbf{G}\mathbf{V}_1 + \mathbf{H}\mathbf{V}_2)\widehat{\mathbf{X}}^{\!\perp } \,]\mathbf{W}^+ + \mathbf{U}_2\mathbf{W}^{\!\perp })\mathbf{y}\right\} \nonumber \\&\quad = [\,\mathbf{F}, \, \mathbf{0}\,]\mathbf{W}^+ \mathbf{V}([\,\mathbf{0}, \, (\mathbf{G}\mathbf{V}_1 + \mathbf{H}\mathbf{V}_2)\widehat{\mathbf{X}}^{\!\perp } \,]\mathbf{W}^+)^{\prime }. \end{aligned}$$
(75)
Applying (60) to (75) and simplifying, we obtain
$$\begin{aligned}&r(Cov\{\,{\mathrm{BLUE}}(\mathbf{F}{\pmb {\alpha }}), \, {\mathrm{BLUP}}(\mathbf{G}{\pmb {\gamma }}+ \mathbf{H}{\pmb {\varepsilon }})\,\})\\&\quad = r\!\left( [\,\mathbf{F}, \, \mathbf{0}\,]\mathbf{W}^+ \mathbf{V}([\,\mathbf{0}, \, (\mathbf{G}\mathbf{V}_1 + \mathbf{H}\mathbf{V}_2)\widehat{\mathbf{X}}^{\!\perp } \,]\mathbf{W}^+)^{\prime } \right) \\&\quad = r\!\left[ \begin{array}{c@{\quad }c@{\quad }c} \mathbf{0}&{} \left[ \begin{array}{c} \widehat{\mathbf{X}}^{\prime }\\ \widehat{\mathbf{X}}^{\!\perp }\mathbf{V}\end{array} \right] &{} \left[ \!\begin{array}{c} \mathbf{0}\\ \widehat{\mathbf{X}}^{\!\perp }(\mathbf{G}\mathbf{V}_1 + \mathbf{H}\mathbf{V}_2)^{\prime } \end{array} \right] \\ {[}\,\widehat{\mathbf{X}}, \, \mathbf{V}\widehat{\mathbf{X}}^{\!\perp }\,] &{} \mathbf{V}&{} \mathbf{0}\\ {[}\,\mathbf{F}, \, \mathbf{0}\,] &{} \mathbf{0}&{} \mathbf{0}\end{array} \right] - 2r(\mathbf{W})\\&\quad = r\!\left[ \begin{array}{c@{\quad }c@{\quad }c} \left[ \begin{array}{c@{\quad }c} \mathbf{0}&{} \mathbf{0}\\ \mathbf{0}&{} -\widehat{\mathbf{X}}^{\!\perp }\mathbf{V}\widehat{\mathbf{X}}^{\!\perp } \end{array} \right] &{} \left[ \begin{array}{c} \widehat{\mathbf{X}}^{\prime } \\ \mathbf{0}\end{array} \right] &{} \left[ \begin{array}{c} \mathbf{0}\\ \widehat{\mathbf{X}}^{\!\perp }(\mathbf{G}\mathbf{V}_1 + \mathbf{H}\mathbf{V}_2)^{\prime } \end{array} \right] \\ {[}\,\widehat{\mathbf{X}}, \, \mathbf{0}\,] &{} \mathbf{V}&{} \mathbf{0}\\ {[}\,\mathbf{F}, \, \mathbf{0}\,] &{} \mathbf{0}&{} \mathbf{0}\end{array}\right] - 2r[\,\widehat{\mathbf{X}}, \, \mathbf{V}\,]\\&\quad = r\!\left[ \begin{array}{c@{\quad }c} \mathbf{0}&{} \widehat{\mathbf{X}}^{\prime } \\ \widehat{\mathbf{X}} &{} \mathbf{V}\\ \mathbf{F}&{} \mathbf{0}\end{array} \right] + r[\, \widehat{\mathbf{X}}^{\!\perp }\mathbf{V}\widehat{\mathbf{X}}^{\!\perp }, \, \widehat{\mathbf{X}}^{\!\perp }(\mathbf{G}\mathbf{V}_1 + \mathbf{H}\mathbf{V}_2)^{\prime } \,] - 2r[\,\widehat{\mathbf{X}}, \, \mathbf{V}\,]\\&\quad = r\!\left[ \begin{array}{c} \widehat{\mathbf{X}} \\ \mathbf{F}\end{array} \right] + r\left[ \begin{array}{c} \widehat{\mathbf{X}}^{\prime } \\ \mathbf{V}\end{array} \right] + r[\, \widehat{\mathbf{X}}, \, \mathbf{V}\widehat{\mathbf{X}}^{\!\perp }, \, (\mathbf{G}\mathbf{V}_1 + \mathbf{H}\mathbf{V}_2)^{\prime } \,] \\&\qquad - \ r(\widehat{\mathbf{X}}) - 2r[\,\widehat{\mathbf{X}}, \, \mathbf{V}\,] \ \ \ \text{(by } \text{(58)) }\\&\quad = r[\, \widehat{\mathbf{X}}, \, \mathbf{V}\widehat{\mathbf{X}}^{\!\perp }, \, \mathbf{V}_1^{\prime } \mathbf{G}^{\prime } +\mathbf{V}_2^{\prime }\mathbf{H}^{\prime } \,] - r[\,\widehat{\mathbf{X}}, \, \mathbf{V}\,] \\&\quad = r[\, \widehat{\mathbf{X}}, \, \mathbf{V}, \, \mathbf{V}_1^{\prime } \mathbf{G}^{\prime } +\mathbf{V}_2^{\prime }\mathbf{H}^{\prime } \,] - r[\,\widehat{\mathbf{X}}, \, \mathbf{V}\,] \ \ \text{(by } \text{ Theorem } \text{1(a)) }\\&\quad = r[\, \widehat{\mathbf{X}}, \, \mathbf{V}, \, \mathbf{0}\,] - r[\,\widehat{\mathbf{X}}, \, \mathbf{V}\,] \ \ \ \text{(by } \text{(5)) },\\&\quad = 0, \end{aligned}$$
which implies that \(Cov\{\,\mathrm{BLUE}(\mathbf{F}{\pmb {\alpha }}), \, \mathrm{BLUP}(\mathbf{G}{\pmb {\gamma }}+ \mathbf{H}{\pmb {\varepsilon }})\,\}\) is a zero matrix, establishing (18). Equation (19) follows from (17) and (18). \(\square \)
Proof of Corollary 2
It is easy to verify by the definition of Moore–Penrose inverse that if \({\fancyscript{R}}(\mathbf{A}) \cap {\fancyscript{R}}(\mathbf{B}) = \{\mathbf{0}\}\), then \([\, \mathbf{A}, \, \mathbf{B}\,]^+ \!= \!{\left[ \!\begin{array}{c} \mathbf{A}^{+}- \mathbf{A}^{+}\mathbf{B}(\mathbf{A}^{\!\perp }\mathbf{B})^+ \\ (\mathbf{A}^{\!\perp }\mathbf{B})^+ \end{array}\!\right] }\!\). From this formula and Theorem 1(a), we obtain
$$\begin{aligned}&\displaystyle \mathbf{W}^+ = [\, \widehat{\mathbf{X}}, \, \mathbf{V}\widehat{\mathbf{X}}^{\!\perp }\,]^+ \! = \!\left[ \!\begin{array}{c} \widehat{\mathbf{X}}^{+} - \widehat{\mathbf{X}}^{+}\mathbf{V}\widehat{\mathbf{X}}^{\!\perp } (\widehat{\mathbf{X}}^{\!\perp }\mathbf{V}\widehat{\mathbf{X}}^{\!\perp })^+\\ (\widehat{\mathbf{X}}^{\!\perp }\mathbf{V}\widehat{\mathbf{X}}^{\!\perp })^+ \end{array}\!\right] \!, \end{aligned}$$
(76)
$$\begin{aligned}&\displaystyle [\mathbf{0}, \, \mathbf{I}_n]\mathbf{W}^+\mathbf{V}\mathbf{W}^+[\mathbf{0}, \, \mathbf{I}_n]^{\prime } = (\widehat{\mathbf{X}}^{\!\perp }\mathbf{V}\widehat{\mathbf{X}}^{\!\perp })^+. \end{aligned}$$
(77)
Results in (a)–(g) follow directly from Theorem 1, Corollary 1, (76), and (77). We also obtain from (12), (76), and (77) that
$$\begin{aligned} Cov\{\,\mathrm{BLUP}(\mathbf{X}{\pmb {\gamma }}), \, \mathrm{BLUP}({\pmb {\varepsilon }}) \,\} = \mathbf{X}\mathbf{V}_1(\widehat{\mathbf{X}}^{\!\perp }\mathbf{V}\widehat{\mathbf{X}}^{\!\perp })^+\mathbf{V}_2^{\prime }. \end{aligned}$$
Applying (59) to the right-hand side of this formula, and simplifying by (56) and (57) yields
$$\begin{aligned}&r(Cov\{\,\mathrm{BLUP}(\mathbf{X}{\pmb {\gamma }}), \, \mathrm{BLUP}({\pmb {\varepsilon }})\,\}) = r\!\left( \mathbf{X}\mathbf{V}_1(\widehat{\mathbf{X}}^{\!\perp } \mathbf{V}\widehat{\mathbf{X}}^{\!\perp })^+\mathbf{V}_2^{\prime } \right) \\&\quad = r\!\left[ \begin{array}{c@{\quad }c} \widehat{\mathbf{X}}^{\!\perp } \mathbf{V}\widehat{\mathbf{X}}^{\!\perp } &{} \widehat{\mathbf{X}}^{\!\perp }\mathbf{V}_2^{\prime }\\ \mathbf{X}\mathbf{V}_1\widehat{\mathbf{X}}^{\!\perp } &{}\mathbf{0}\end{array} \right] - r(\widehat{\mathbf{X}}^{\!\perp }\mathbf{V}\widehat{\mathbf{X}}^{\!\perp }) = r\!\left[ \begin{array}{c@{\quad }c@{\quad }c} \mathbf{V}&{} \widehat{\mathbf{X}} &{} \mathbf{V}_2^{\prime }\\ \widehat{\mathbf{X}}^{\prime } &{}\mathbf{0}&{}\mathbf{0}\\ \mathbf{X}\mathbf{V}_1 &{}\mathbf{0}&{}\mathbf{0}\end{array}\right] - r\!\left[ \begin{array}{c@{\quad }c} \mathbf{V}&{} \widehat{\mathbf{X}}\\ \widehat{\mathbf{X}}^{\prime } &{} \mathbf{0}\end{array}\right] \!, \end{aligned}$$
establishing (45). Setting the right-hand side of (45) equal to zero leads to the equivalence of (i), (ii), and (iii) in (h).\(\square \)
