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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.

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  • $\begingroup$ 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? $\endgroup$
    – whuber
    Jul 19 at 13:44

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