Compressing covariance matrix from 3X3 to 2X2

Question

thanks for answering my question. My question is,

Let $v_3=[a,b,c]^T$ is a probabilistic 3-D vector variable which is distributed by zero mean Gaussian of dense covariance matrix $\Sigma_{3\times3}$. Let's assume we could expect the value of an element of the vector by other elements with some uncertainty, for example, $c=f(a,b)+w$, where $f$ is some predefined deterministic function and $w \sim \mathcal{N}(0,\sigma)$ is zero mean Gaussian of covariance $\sigma$. Then, is it able to "compress" the vector and its covariance by 2-D without loss of any information? I mean, is it possible to explicitly express the 2-by-2 covariance matrix of $v_2=[a,b]^T$ by using $\Sigma_{3\times3}$, $f$ and $\sigma$?

I'm not sure whether the specification of the form of $f$ is necessary to solve the problem, but at least in my application, $f$ might take a form of simple linear relationship (eg. c = 3a+5b + w).

Answer 1

The $2\times2$ covariance matrix of $v_2=[a,b]^T$ is the matrix obtained from $\Sigma_{3\times3}$ by removing the 3rd row and the 3rd column of $\Sigma_{3\times3}$.