# Covariance of an uncertain vector going through an uncertain rotation

I have a vector $$\omega$$ and it associated covariance, a rotation $$\mathbf R$$ and its associated covariance. What is the covariance of $$\mathbf R \cdot \omega$$ ?

More rigorously: I have an estimation of a vector expressed in the $$A$$ frame: $$^A \mathbf \omega \in \mathbb R^3$$, and its associated covariance $$\Sigma_{^A\omega} \in \mathbb R^{3\times3}$$. I also have an estimate of the rotation between $$A$$ and $$N$$ expressed by the vector $$\mathbf \theta \in \mathfrak{so}(3)$$ such that the rotation matrix from $$A$$ to $$N$$ is: $$^N \mathbf R _{A} = \text{expm}(\theta) \in SO(3)$$. This estimate of the rotation vector $$\mathbf \theta$$ has the associated covariance $$\Sigma_{\theta} \in \mathbb R^{3\times3}$$.

Given that $$^{N}\omega= \;^{N} \mathbf{R}_{A} \cdot \,^{A} \mathbf \omega \;$$, what is $$\Sigma_{^N\omega}$$ ?

Thank you.

[EDIT]: If the rotation is without covariance (fully known), then $$\Sigma_{^N\omega} = \, ^N \mathbf R _{A} \cdot \Sigma_{^A\omega} \cdot ^N \mathbf R _{A}^\intercal$$

On a general basis, given a function $$f: \mathbb{R}^m\to \mathbb{R}^n$$ and a vector $$\mathbf x \in \mathbb R^m$$ and its associated covariance $$\Sigma_x \in \mathbb R^{m \times m}$$, if $$\mathbf y = f(\mathbf x)$$ then:
$$\Sigma_x \simeq \left . \frac{\partial f}{\partial \mathbf x}\right |_{\mathbf x} \Sigma_x \left . \frac{\partial f}{\partial \mathbf x}\right |_{\mathbf x} ^\intercal$$