# Vector spaces equation

1. Aug 2, 2013

### GoodSpirit

Hi everyone,

I have the following term:

$$S=dim(R( \begin{bmatrix} \Sigma_{x1}\\ CC^{T}\\ \end{bmatrix} \phi_1) \cap R( \begin{bmatrix} CC\\ \Sigma_{x2} \\ \end{bmatrix} \phi_2) )$$

where

$$\Sigma_{x1} \in \mathbb{R}^{nxn} \\ \Sigma_{x2} \in \mathbb{R}^{nxn}\\ CC \in \mathbb{R}^{nxn}\\ \phi_1 \in \mathbb{R}^{l1xn}\\ \phi_2 \in \mathbb{R}^{l2xn}\\$$

and

$$rk(\Sigma_{x1})=s1\\ rk(\Sigma_{x2})=s2$$

$$R(X)$$ is the column space (range) defined by the columns of the matrix X.

My goal is to express S in terms of s1, s2, l1 and l2.
I really thank you for your help.

All the best

Ricardo Sousa