# Solved – Propagation of uncertainty through a linear system of equations

If I have a system of equations, \$Ax=B\$ where the elements of \$B\$ have been experimentally determined and as such each element has some uncertainty, how would I propagate this to the elements of \$x\$?

\$\$
left[begin{matrix}
a_{11} & a_{12}\
a_{21} & a_{22}\
end{matrix}right]
left[begin{matrix} x_{11}\x_{21}end{matrix}right]=
left[begin{matrix} b_{11}pmsigma_{b_{11}}\b_{21}pmsigma_{b_{21}}end{matrix}right]
\$\$

For instance, in a system like the one above, how do I account for the error in \$B\$ when solving for \$x\$? I am trying to find
\$sigma_{x_{11}}\$ and \$sigma_{x_{12}}\$.

Contents

Let me translate into statistician. So \$B\$ is a random variable where \$B = beta + varepsilon\$, with \$text{Var}(varepsilon)\$ = \$Sigma_B\$, for \$Sigma_B\$ known. An observation is taken, and the observed value of \$B\$ is \$b\$.

Assuming \$A\$ is invertible, the solution of \$Ax = b\$ is \$A^{-1}b\$. Let \$C = A^{-1}\$ for the moment.

\$text{Var}(Cb) = C,text{Var}(b),C^top = A^{-1} Sigma_B (A^{-1})^top\$

If the two components of \$b\$ are independent, then \$Sigma_B\$ is diagonal, with diagonal the squares of your \$sigma\$'s. That variance-covariance matrix of \$x\$ is in general not diagonal, meaning the values are correlated. The square roots of the diagonal elements of \$ A^{-1} Sigma_B (A^{-1})^top\$ are the standard deviations of the components of \$x\$.

This approach applies to more than two dimensions as well.

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