Here are just some simple notes about these 2 equivalent models

#### MME for GBLUP

You can understand GBLUP is a “improved” version of traditional PBLUP

$\left[ \begin{array}{cc} \mathbf{X'R}^{-1}\mathbf{X} & \mathbf{X'R}^{-1}\mathbf{Z}\\ \mathbf{Z'R}^{-1}\mathbf{X} & \mathbf{Z'R}^{-1}\mathbf{Z}+(\mathbf{G}\sigma^2_a)^{-1} \end{array}\right] \left[ \begin{array}{c} \hat{\mathbf{b}}\\ \hat{\mathbf{u}} \end{array}\right] = \left[ \begin{array}{c} \mathbf{X'R}^{-1}\mathbf{y}\\ \mathbf{Z'R}^{-1}\mathbf{y} \end{array}\right]$ $\left[ \begin{array}{cc} \mathbf{X'}\mathbf{X} & \mathbf{X'}\mathbf{Z}\\ \mathbf{Z'}\mathbf{X} & \mathbf{Z'}\mathbf{Z}+\mathbf{G}^{-1}\lambda \end{array}\right] \left[ \begin{array}{c} \hat{\mathbf{b}}\\ \hat{\mathbf{u}} \end{array}\right] = \left[ \begin{array}{c} \mathbf{X'}\mathbf{y}\\ \mathbf{Z'}\mathbf{y} \end{array}\right]$

where $\sigma^2_a$ is the total genetic variance, $\lambda=\sigma^2_e/\sigma^2_a$
$\mathbf{u}\;\sim\;N(\mathbf{0},\mathbf{G}\sigma^2_a)$

#### Model for SNP-BLUP

The marker effects $\mathbf{g}_i$ were assumed to be identically and independently distributed

$\mathbf{y=1\mu + Mg + e}$

where
$\mathbf{g}\;\sim\;N(\mathbf{0},\mathbf{I}\sigma^2_g)$
$\mathbf{e}\;\sim\;N(\mathbf{0},\mathbf{D}\sigma^2_e)$

where
$d_{ii} = 1/\omega_i$
$\omega_i = EDC_i/\lambda$

MME

$\left[ \begin{array}{cc} \mathbf{X'R}^{-1}\mathbf{X} & \mathbf{X'R}^{-1}\mathbf{Z}\\ \mathbf{Z'R}^{-1}\mathbf{X} & \mathbf{Z'R}^{-1}\mathbf{Z}+(\mathbf{I}\sigma^2_g)^{-1} \end{array}\right] \left[ \begin{array}{c} \hat{\mathbf{b}}\\ \hat{\mathbf{a}} \end{array}\right] = \left[ \begin{array}{c} \mathbf{X'R}^{-1}\mathbf{y}\\ \mathbf{Z'R}^{-1}\mathbf{y} \end{array}\right]$ $\left[ \begin{array}{cc} \mathbf{X'}\mathbf{X} & \mathbf{X'}\mathbf{Z}\\ \mathbf{Z'}\mathbf{X} & \mathbf{Z'}\mathbf{Z}+\mathbf{I}\lambda \end{array}\right] \left[ \begin{array}{c} \hat{\mathbf{b}}\\ \hat{\mathbf{a}} \end{array}\right] = \left[ \begin{array}{c} \mathbf{X'}\mathbf{y}\\ \mathbf{Z'}\mathbf{y} \end{array}\right]$

where
$\sigma^2_g$ is the SNP variance for each SNP
$\lambda=\sigma^2_e/\sigma^2_g$
$\sigma^2_a=2\sum_{i=1}^mp_iq_i\sigma^2_g$

The DGV

$\mathbf{\hat{a} = 1\hat{\mu} + M\hat{g}}$