# How to simplify fallowing martrices' Expected value elements in the equation

I have a matrix equation with four variables inside, $$x^1_{00}$$, $$x^1_{tt}$$ and $$x^2_{tt}$$, $$x^2_{tt}$$. $$x^1_{00}$$, $$x^1_{tt}$$ come from the same distribution , they are only shifted by timelag $$t$$. The same goes for $$x^2_{tt}$$, $$x^2_{tt}$$. Also both are mean-free (their mean is zero).

I have a matrix equation as fallowing $$\begin{bmatrix} E[x^1_{00} x^1_{00}] & E[x^1_{00} x^2_{00}] \\ E[x^2_{00} x^1_{00}] & E[x_{00}^2 x^2_{00}] \end{bmatrix}^{-1/2} \begin{bmatrix} E[x^1_{00} x^1_{tt}] & E[x^1_{00} x^2_{tt}] \\ E[x^2_{00} x^1_{tt}] & E[x_{00}^2 x^2_{tt}] \end{bmatrix} \begin{bmatrix} E[x^1_{tt} x^1_{tt}] & E[x^1_{tt} x^2_{tt}] \\ E[x^2_{tt} x^1_{tt}] & E[x_{tt}^2 x^2_{tt}] \end{bmatrix}^{-1/2}$$

What I want to achieve is to simplify elements inside the matrices: e.g. $$E[x^1_{tt} x^1_{tt}]$$ and $$E[x^1_{00} x^1_{00}]$$ can be noted as $$\sigma^2(x^1_{00})$$, and the same goes for $$x^1_{00}$$ so:

$$\begin{bmatrix} \sigma^2(x^1_{00}) & E[x^1_{00} x^2_{00}] \\ E[x^2_{00} x^1_{00}] & \sigma^2(x^2_{00}) \end{bmatrix}^{-1/2} \begin{bmatrix} E[x^1_{00} x^1_{tt}] & E[x^1_{00} x^2_{tt}] \\ E[x^2_{00} x^1_{tt}] & E[x_{00}^2 x^2_{tt}] \end{bmatrix} \begin{bmatrix} \sigma^2(x^1_{00}) & E[x^1_{tt} x^2_{tt}] \\ E[x^2_{tt} x^1_{tt}] & \sigma^2(x^2_{00}) \end{bmatrix}^{-1/2}$$

But the rest is quite hard for me to simplify and can't identify with high certainty elements, that could be reduced to the same variable.

• Could you explain why you duplicate every subscript in your notation? I suspect if you were to disclose the statistical application, it would help all readers interpret the meaning of your formula and thereby suggest a (huge) simplification. Incidentally, which square roots of these matrices do you have in mind? The usual symmetric ones?
– whuber
Jul 19 at 13:44