Let $I=\begin{pmatrix}I_1\\\vdots\\ I_n \end{pmatrix}$ and $X=\begin{pmatrix}X_1\\\vdots\\ X_p \end{pmatrix}$ be two random vectors and $\Omega_I$ a random variable. I am looking for A such that:
$$\underbrace{\frac{1}{V(\Omega_I)}}_{Scalar}\underbrace{Cov(X,I)}_{(p,n)}\underbrace{V(I)^{-1}}_{(n,n)}\underbrace{Cov(I,\Omega_I)}_{(n,1)}=A \underbrace{\frac{1}{V(\Omega_I)}}_{Scalar}\underbrace{Cov(X,\Omega_I)}_{(p,1)}$$
When X et I are random variables, $A=\frac{\rho_{XI}\rho_{I\Omega_I}}{\rho_{X\Omega_I}}$.
I have the intuition that this A (which might be a scalar) should be able to the same result but with $\rho$ the coefficient of multiple correlations (Person's R). However, I do not manage to show it?
Would anyone know how to solve it?
Thank you very much.