I have an equation
T3=T1*T2
where T is 3*3 Transformation matrix representing position of an object in 3D. Now each of these position has some error in the form of 3*3 covariance matrix i-e ∑.
My question is Covariance of T3 is the simple multiplication of covariance of T1 and T2 or is there any complex mathematics involved in it? i-e ∑3 = ∑1*∑2
Any idea or suggestions.
Clearly the main task here is to reach a precise mathematical question. So, let us assume that the setting is that one starts from some vector $U_1$ in $\mathbb R^3$, that one first transforms $U_1$ into an intermediate vector $U_2$ in $\mathbb R^3$ using $T_1$ and some noise, thus $$U_2=T_1U_1+W_1$$ where $W_1$ has mean zero and covariance matrix $\Sigma_1$, and that one then transforms $U_2$ into a final vector $U_3$ using $T_2$ and some noise, thus $$U_3=T_2U_2+W_2$$ where $W_2$ is independent of $W_1$ and has mean zero and covariance matrix $\Sigma_2$. And the question is to determine the covariance matrix $\Sigma_3$ of $U_3$.
To do so, note that $$U_3=T_2T_1U_1+W_3\qquad\text{where}\qquad W_3=T_2W_1+W_2$$ hence $W_3$ is centered and $$\Sigma_3=E(W_3W_3^*)=T_2E(W_1W_1^*)T_2^*+E(W_2W_2^*),$$ that is,
Already in the scalar case, it is clear that the noise $\Sigma_1$ of the first step must be scaled by the second transform $T_2$, thus such a formula should not come as a surprise.