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Why are these two similar time series identical?
I'm having trouble explaining myself on this one:
Given $y_1, y_2 \in \mathbb{R}^2$, $\sigma^2 > 0$, and $\rho = 1$, why are the following two random vectors identical?
$
y_1 = \begin{bmatrix} \sigma \rho x_1 \\ \sigma (1- \rho) x_2 \end{bmatrix}, \quad
y_2 = \begin{bmatrix} \sigma (1- \rho) x_1 \\ \sigma \rho x_2 \end{bmatrix}?
$
I understand that $\mathbb{E}(y_1) = \mathbb{E}(y_2)$, but I don't get why they're equal to $\sigma^2$ times any constant times $x$, where $x$ is an arbitrary vector in $\mathbb{R}^2$.
For context, $x_1$ and $x_2$ are bivariate normal distributed $N(0, \Sigma)$ where $\Sigma = \begin{pmatrix} 1 & \rho \\ \rho & 1 \end{pmatrix}$.
A:
Note that
\begin{align*}
\mathbf{y_1}&=\sigma\left(\rho\mathbf{x_1}+\left(1-\rho\right)\mathbf{x_2}\right)\qquad\text{and}\\
\mathbf{y_2}&=\sigma\left(1-\rho\right)\mathbf{x_1}+\sigma\rho\mathbf{x_2}
\end{align*}
The expectation of the first vector is
$
\mathbb{E}[\mathbf{y_1}]=\sigma\left(\rho\mathbb{E}\left 0b46394aab
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