Suppose I have a covariance matrix of a random vector $$X=\begin{bmatrix}X1\\ X2\\ X3\\ X4 \end{bmatrix}$$

$$C_X=\begin{bmatrix}a&b&c&d\\ e&f&g&h\\ i&j&k&l\\ m&n&o&p \end{bmatrix}$$

where $X1,X2,X3,X4$ are random variables and $a-p$ are just real constants.

How do I partition $X$ into two sub vectors such that one of the sub-vectors has a non-singular covariance and other sub-vector is a linear transformation of the first?

Are there are list of steps I can follow to accomplish this?


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