$\sigma_{x_1x_2}$ is known as the covariance between $x_1$ and $x_2$

For a function with n independent variables, the errors propagate as:

$\Sigma_{zz}= \left[ \begin{matrix} a_1 & a_2 & \cdots & a_n \end{matrix} \right] \left[ \begin{matrix} \sigma_{x_1}^{2} & \sigma_{x_1x_2} & \cdots & \sigma_{x_1x_n}\\ \sigma_{x_2x_1} & \sigma_{x_2}^{2} & \cdots & \sigma_{x_2x_n}\\ \vdots & \vdots & \ddots & \vdots\\ \sigma_{x_nx_1} & \sigma_{x_nx_2} & \cdots & \sigma_{x_n}^{2} \end{matrix} \right] \left[ \begin{matrix} a_1\\ a_2\\ \vdots\\ a_n \end{matrix} \right]$

Given, standard deviations/precisions of $x_1$ and $x_2$ given, standard deviation or precision of the derived quantity $z$ can be calculated in the matrix form as follows…

$z=AX$, where A =$\left[ \begin{matrix} a_1 & a_2 \end{matrix} \right]$. $X$ will be equal to the matrix $\left[ \begin{matrix} x_1\\ x_2\\ \end{matrix} \right]$.

We know $\sigma_z^{2}=$A × $\Sigma$x × A^T, or…

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with

• $\left[ \begin{matrix} a_1 & a_2 & \cdots & a_n \end{matrix} \right]$ = A
• $\left[ \begin{matrix} \sigma_{x_1}^{2} & \sigma_{x_1x_2} & \cdots & \sigma_{x_1x_n}\\ \sigma_{x_2x_1} & \sigma_{x_2}^{2} & \cdots & \sigma_{x_2x_n}\\ \vdots & \vdots & \ddots & \vdots\\ \sigma_{x_nx_1} & \sigma_{x_nx_2} & \cdots & \sigma_{x_n}^{2} \end{matrix} \right]$ = $\Sigma$x
• $\left[ \begin{matrix} a_1\\ a_2\\ \vdots\\ a_n \end{matrix} \right]$ = A^T