first time to use this website, hope I made it correctly.
Suppose $\underline{X}\sim n(\underline{\mu},\Sigma)$ where $\Sigma= (\begin{matrix}&\Sigma_{11} &\Sigma_{12}\\&\Sigma_{21} & \Sigma_{22}\end{matrix})$where $\Sigma_{11}$ is (n-1)*(n-1) dimensional, $\Sigma_{12}$ is (n-1)1 dimensional, $\Sigma_{21}$ is 1(n-1) dimensional, $\Sigma_{22}$ is 1*1 dimensional. Now we already know that $X_1 = x_1, X_2 = x_2, \cdots, X_{n-1} = x_{n-1}$, thus the minimum MSE of $X_n$ is $\hat{X_n} = E[X_n|X_1 = x_1, X_2 = x_2, \cdots, X_{n-1} = x_{n-1}] = \mu_n + \Sigma_{21}\Sigma_{11}^{-1}(\begin{matrix}&x_1-\mu_1\\ &x_2-\mu_2\\ &\cdots\\ &x_{n-1}-\mu_{n-1})\end{matrix}$.
My question is: how to calculate the corresponding MSE? I'm not very familiar with the matrix calculation. enter image description here
