在基因组选择模型介绍下提到了H逆矩阵，这里是补上证明。

H逆证明

首先，按照无基因型个体在前，有基因型个体在后的顺序，H 阵的公式为 \[ \begin{aligned} \mathbf{H} &=\left(\begin{array}{cc} \operatorname{var}\left(\mathbf{u}_{1}\right) & \operatorname{cov}\left(\mathbf{u}_{1}, \mathbf{u}_{2}\right) \\ \operatorname{cov}\left(\mathbf{u}_{2}, \mathbf{u}_{1}\right) & \operatorname{var}\left(\mathbf{u}_{2}\right) \end{array}\right) \\ &=\left(\begin{array}{cc} \mathbf{A}_{11}+\mathbf{A}_{12} \mathbf{A}_{22}^{-1}\left(\mathbf{G}-\mathbf{A}_{22}\right) \mathbf{A}_{22}^{-1} \mathbf{A}_{21} & \mathbf{A}_{12} \mathbf{A}_{22}^{-1} \mathbf{G} \\ \mathbf{G} \mathbf{A}_{22}^{-1} \mathbf{A}_{21} & \mathbf{G} \end{array}\right) \\ &=\mathbf{A}+\left[\begin{array}{cc} \mathbf{A}_{12} \mathbf{A}_{22}^{-1}\left(\mathbf{G}-\mathbf{A}_{22}\right) \mathbf{A}_{22}^{-1} \mathbf{A}_{21} & \mathbf{A}_{12} \mathbf{A}_{22}^{-1}\left(\mathbf{G}-\mathbf{A}_{22}\right) \\ \left(\mathbf{G}-\mathbf{A}_{22}\right) \mathbf{A}_{22}^{-1} \mathbf{A}_{21} & \mathbf{G}-\mathbf{A}_{22} \end{array}\right] \end{aligned} \] 如果我们按照有基因型个体在前，无基因型个体在后的顺序，我们有 \[ \begin{aligned} \mathbf{H} &=\left(\begin{array}{cc} \operatorname{var}\left(\mathbf{u}_{1}\right) & \operatorname{cov}\left(\mathbf{u}_{1}, \mathbf{u}_{2}\right) \\ \operatorname{cov}\left(\mathbf{u}_{2}, \mathbf{u}_{1}\right) & \operatorname{var}\left(\mathbf{u}_{2}\right) \end{array}\right) \\ &=\left(\begin{array}{cc} \mathbf{G} & \mathbf{G} \mathbf{A}_{11}^{-1} \mathbf{A}_{12} \\ \mathbf{A}_{21} \mathbf{A}_{11}^{-1} \mathbf{G} & \mathbf{A}_{22}+\mathbf{A}_{21} \mathbf{A}_{11}^{-1}\left(\mathbf{G}-\mathbf{A}_{11}\right) \mathbf{A}_{11}^{-1} \mathbf{A}_{12} \end{array}\right) \\ \end{aligned} \] 相应的 \(\mathbf{A}\) 阵分块为 \[ \mathbf{A} = \left(\begin{array}{cc} \mathbf{A}_{11} & \mathbf{A}_{12} \\ \mathbf{A}_{21} & \mathbf{A}_{22} \\ \end{array}\right) \] 根据分块矩阵求逆的一般规则 (分块矩阵求逆公式 ，如果 \(\mathbf{A}_{11}\) 和 \(\mathbf{A}_{22}-\mathbf{A}_{21} \mathbf{A}_{11}^{-1} \mathbf{A}_{12}\) 可逆，我们有 \[ \mathbf{A}^{-1}=\left[\begin{array}{cc} \mathbf{A}_{11}^{-1}+\mathbf{A}_{11}^{-1} \mathbf{A}_{12}\left(\mathbf{A}_{22}-\mathbf{A}_{21} \mathbf{A}_{11}^{-1} \mathbf{A}_{12}\right)^{-1} \mathbf{A}_{21} \mathbf{A}_{11}^{-1} & -\mathbf{A}_{11}^{-1} \mathbf{A}_{12}\left(\mathbf{A}_{22}-\mathbf{A}_{21} \mathbf{A}_{11}^{-1} \mathbf{A}_{12}\right)^{-1} \\ -\left(\mathbf{A}_{22}-\mathbf{A}_{21} \mathbf{A}_{11}^{-1} \mathbf{A}_{12}\right)^{-1} \mathbf{A}_{21} \mathbf{A}_{11}^{-1} & \left(\mathbf{A}_{22}-\mathbf{A}_{21} \mathbf{A}_{11}^{-1} \mathbf{A}_{12}\right)^{-1} \end{array}\right] \] 同样地，对于 \(\mathbf{H}\) 矩阵，应用相同的规则，化简后得到 \[ \begin{aligned} &\mathbf{H}^{-1} \\ &=\left[\begin{array}{cc} \mathbf{G}^{-1}+\mathbf{A}_{11}^{-1} \mathbf{A}_{12}\left(\mathbf{A}_{22}-\mathbf{A}_{21} \mathbf{A}_{11}^{-1} \mathbf{A}_{12}\right)^{-1} \mathbf{A}_{21} \mathbf{A}_{11}^{-1} & -\mathbf{A}_{11}^{-1} \mathbf{A}_{12}\left(\mathbf{A}_{22}-\mathbf{A}_{21} \mathbf{A}_{11}^{-1} \mathbf{A}_{12}\right)^{-1} \\ -\left(\mathbf{A}_{22}-\mathbf{A}_{21} \mathbf{A}_{11}^{-1} \mathbf{A}_{12}\right)^{-1} \mathbf{A}_{21} \mathbf{A}_{11}^{-1} & \left(\mathbf{A}_{22}-\mathbf{A}_{21} \mathbf{A}_{11}^{-1} \mathbf{A}_{12}\right)^{-1} \end{array}\right] \\ &=\left[\begin{array}{cc} \mathbf{G}^{-1}-\mathbf{A}_{11}^{-1} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{array}\right]+\mathbf{A}^{-1} \end{aligned} \]

参考文献

1. Christensen O F, Lund M S. Genomic prediction when some animals are not genotyped[J]. Genetics Selection Evolution, 2010, 42(1): 1-8.
2. Aguilar I, Misztal I, Johnson D L, et al. Hot topic: a unified approach to utilize phenotypic, full pedigree, and genomic information for genetic evaluation of Holstein final score[J]. Journal of dairy science, 2010, 93(2): 743-752.