Let $ X $ be a $p$-dimensional random vector, $X^{T} = \left (x_{1}, \ldots, x_{p} \right) $, with $ E(X)=0$ and $Variance(X)=V $. Suppose you are interested in minimizing $$ E\left[\left(x_{1}-\sum_{j \neq 1}^{p} b_{1 j} x_{j}\right)^{2}\right]. $$
Furthermore, suppose that you are not only interested in expressing $ x_ {1} $ in terms of the rest of the variables (as in the previous expression), but that we want to simultaneously express each one in terms of the others. That is, consider the problem of minimizing, with respect to $B=(b_{ij})$ $$ L=E\left[\sum_{i=1}^{p}\left(x_{i}-\sum_{j \neq i}^{p} b_{i j} x_{j}\right)^{2}\right]=E\left[\|X-B X\|^{2} \right]. $$ Here $\|\cdot\|^2$ is the Euclidean norm. Show the second equality $ and $ show that $L=\left\|(I-B) V^{1 / 2}\right\|_{F}^{2}$, where $\|\cdot\|_{F}$ is the Frobenius norm.
Hint: if $z$ is a random vector with mean $\mu$ and variance $\Sigma$ then $E\left(z^{T} A z\right)=\mu^{T} A \mu+\operatorname{tr}(A \Sigma)$.
Attempt of solution:
Starting from $E\left[\|X-B X\|^{2} \right]$ we have
$$ E\left[ \left\| \left[\begin{array}{c} x_{1} \\ x_{2} \\ \vdots \\ x_{p} \end{array}\right] - \left[\begin{array}{cccc} b_{11} & b_{12} & \ldots & b_{1 n} \\ b_{21} & b_{22} & \ldots & b_{2 n} \\ \vdots & \vdots & \vdots & \vdots \\ b_{p 1} & b_{p 2} & \ldots & b_{p p} \end{array}\right]\left[\begin{array}{c} x_{1} \\ x_{2} \\ \vdots \\ x_{p} \end{array}\right]\right\|^2 \right]= E\left[\left\|\left[\begin{array}{c} x_1 - \sum_{j=1}^{p} b_{1 j} x_{j} \\ x_2 - \sum_{j=1}^{p} b_{2 j} x_{j} \\ \vdots \\ x_p - \sum_{j=1}^{p} b_{p j} x_{j} \end{array}\right]\right\|^2 \right]= E\left[\sum_{i=1}^{p}\left(x_{i}-\sum_{j = 1}^{p} b_{i j} x_{j}\right)^{2}\right]. $$ But $$ \mathbf{E}\left[\sum_{i=1}^{p}\left(x_{i}-\sum_{j \neq i}^{p} b_{i j} x_{j}\right)^{2}\right] \neq E\left[\sum_{i=1}^{p}\left(x_{i}-\sum_{j = 1}^{p} b_{i j} x_{j}\right)^{2}\right]. $$
Do you have any ideas or suggestions? Is there a bibliography where you can consult these topics?