Let $[v_1(k) \, v_2(k)]^T\in\mathbb{R}^2$ be a stochastic process that is independent of the process $\beta_i(k)$ for all $i=1,2,3,4$, with $\mathbb{E}[\beta_i(k)]=\bar\beta_i $. Both of them are assumed to be stationary processes. Does the following hold:
$\mathbb{E}\left[[v_1(k) \, v_2(k)] \left[\begin{array}{cc}\beta_1(k) & \beta_2(k)\\ \beta_3(k) & \beta_4(k)\end{array}\right]\left[\begin{array}{c}v_1(k)\\v_2(k)\end{array}\right]\right]=\mathbb{E}\left[[v_1(k) \, v_2(k)] \left[\begin{array}{cc}\bar\beta_1 & \bar\beta_2\\ \bar\beta_3 & \bar\beta_4\end{array}\right]\left[\begin{array}{c}v_1(k)\\v_2(k)\end{array}\right]\right]$
For $\boldsymbol{v}(k)^T:=[v_1(k) \, v_2(k)]$ , $\boldsymbol{\beta}(k):=\left[\begin{array}{cc}\beta_1(k) & \beta_2(k)\\ \beta_3(k) & \beta_4(k)\end{array}\right]$ use the conditional expectation:
$\mathbb{E}[\boldsymbol{v}(k)^T \boldsymbol{\beta}(k)\boldsymbol{v}(k)]=\mathbb{E}\left[\mathbb{E}\left[\boldsymbol{v}(k)^T \boldsymbol{\beta}(k)\boldsymbol{v}(k)\quad|\boldsymbol{v}(k)\right]\right]$
$=\mathbb{E}\left[\boldsymbol{v}(k)^T \mathbb{E}\left[\boldsymbol{\beta}(k)|\boldsymbol{v}(k)\right]\boldsymbol{v}(k)\right]=\mathbb{E}\left[\boldsymbol{v}(k)^T \mathbb{E}\left[\boldsymbol{\beta}(k)\right]\boldsymbol{v}(k)\right]$ where I used independence of $\boldsymbol{v}(k)$ and $\boldsymbol{\beta}(k)$ in the last step. The other equations hold as basic properties of the conditional expectation.