I tried to get the product of Multivariate Gausssian pdf.
In 8.1.8, this says \begin{align*} G(x-v_i, \Sigma_i) \times G(x-v_j, \Sigma_j) = G(v_i-v_j, \Sigma_i+\Sigma_j) G(x-v, \Sigma) \\ = \frac{1}{(2\pi)^{1/2} |\Sigma_i+\Sigma_j|^{1/2}} \exp\left\{ -\frac{1}{2} (v_i - v_j)^T (\Sigma_i+\Sigma_j)^{-1} (v_i - v_j) \right\} G(x-v, \Sigma) \end{align*}
However, my answer is as follows:
At first, suppose that \begin{align} G(x-v_i, \Sigma_i) &= \frac{1}{(2\pi)^{d/2} |\Sigma_i|^{1/2}} \exp\left\{ -\frac{1}{2} (x - v_i)^T (\Sigma_i)^{-1} (x - v_i) \right\} \\ &= \exp\left\{ -\frac{d}{2}\log(2\pi) -\frac{1}{2}\log|\Sigma_i| -\frac{1}{2} (x - v_i)^T (\Sigma_i)^{-1} (x - v_i) \right\} \\ &= \exp\left\{ -\frac{1}{2} \left[ d\log(2\pi) +\log|\Sigma_i| + (x - v_i)^T (\Sigma_i)^{-1} (x - v_i) \right] \right\} \\ &= \exp\left\{ -\frac{1}{2} \left[ d\log(2\pi) +\log|\Sigma_i| + x^T \Sigma_i^{-1}x - 2x^T\Sigma_i^{-1}v_i + v_i^T\Sigma_i^{-1}v_i \right] \right\} \\ &= \exp\left\{ -\frac{1}{2} x^T \Sigma_i^{-1}x + x^T\Sigma_i^{-1}v_i -\frac{1}{2} \left[ d\log(2\pi) +\log|\Sigma_i| + v_i^T\Sigma_i^{-1}v_i \right] \right\} \\ &= \exp\left\{ -\frac{1}{2} x^T \Sigma_i^{-1}x + x^T\Sigma_i^{-1}v_i \right\} \exp\left\{ -\frac{1}{2} \left[ d\log(2\pi) +\log|\Sigma_i| + v_i^T\Sigma_i^{-1}v_i \right] \right\} \\ &= \exp\left\{ -\frac{1}{2} x^T \Sigma_i^{-1}x + x^T\eta_i \right\} \exp\left\{ -\frac{1}{2} \left[ d\log(2\pi) +\log|\Sigma_i| + \eta_i^T\Sigma_i\eta_i \right] \right\} \\ &= \exp\left\{ -\frac{1}{2} x^T \Sigma_i^{-1}x + x^T\eta_i + \zeta_i \right\} \end{align} where \begin{align} \eta_i = \Sigma_i^{-1} v_i \\ \zeta_i = -\frac{1}{2} \left[ d\log(2\pi) +\log|\Sigma_i| + \eta_i^T\Sigma_i\eta_i \right] \end{align}
Let's calculate
\begin{align} G(x-v_i, \Sigma_i) \times G(x-v_j, \Sigma_j) \end{align}
\begin{align} &= \exp\left\{ -\frac{1}{2} x^T \Sigma_i^{-1}x + x^T\eta_i + \zeta_i \right\} \exp\left\{ -\frac{1}{2} x^T \Sigma_j^{-1}x + x^T\eta_j + \zeta_j \right\} \\ &= \exp\left\{ -\frac{1}{2} x^T (\Sigma_i^{-1}+\Sigma_j^{-1})x + x^T(\eta_i+\eta_j) + \zeta_i + \zeta_j \right\} \\ &= \exp\left\{ -\frac{1}{2} x^T \Sigma^{-1}x + x^T\eta + \zeta - \zeta + \zeta_i + \zeta_j \right\} \\ &= \exp\left\{ -\frac{1}{2} x^T \Sigma^{-1}x + x^T\eta + \zeta \right\} \exp\left\{ - \zeta + \zeta_i + \zeta_j \right\} \\ &= G(x-v, \Sigma) \exp\left\{ - \zeta + \zeta_i + \zeta_j \right\} \end{align} where \begin{align} \Sigma^{-1} = \Sigma_i^{-1}+\Sigma_j^{-1} \\ \eta = \eta_i+\eta_j \\ \zeta = -\frac{1}{2} \left[ d\log(2\pi) +\log|\Sigma| + \eta^T\Sigma\eta \right] \end{align}
I could not derive the following fact: \begin{align} \exp\left\{ - \zeta + \zeta_i + \zeta_j \right\} = \frac{1}{(2\pi)^{1/2} |\Sigma_i+\Sigma_j|^{1/2}} \exp\left\{ -\frac{1}{2} (v_i - v_j)^T (\Sigma_i+\Sigma_j)^{-1} (v_i - v_j) \right\} \end{align}
Where did I make a mistake?
Thank you in advance.