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As the title states. Is there any significance behind this matrix? Is there a name?

Does it give any information that is not present in the first matrix?

mathplzfun
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  • Sometimes the pseudo-inverse matrix is useful when the original matrix does not have full rank, i.e., not invertible – Sandipan Dey Sep 11 '21 at 22:56
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    The covariance matrix is symmetric, so why did you use $X'$? – Firebug Sep 11 '21 at 23:12
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    It might be worth mentioning what led you to even consider the expression $(\Sigma^\top\Sigma)^{-1}=\Sigma^{-2}$ as a quantity of interest; for $X$ being a *design matrix* (including the constant) in regression the expression $(X^\top X)^{-1}$ certainly has an interpretation, since that's proportional to the covariance matrix of the parameters. – Glen_b Sep 12 '21 at 01:19
  • See https://stats.stackexchange.com/questions/364554/intuition-behind-xtx-1-in-closed-form-of-w-in-linear-regression for X as design matrix instead of the covariance matrix. – Sextus Empiricus Sep 12 '21 at 17:38

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If $\mathbf{x}$ is a covariance matrix then $(\mathbf{x}^\text{T} \mathbf{x})^{-1}$ is just a big mess with no particularly useful meaning (so far as I'm aware). However, you might be interested to note that if $\mathbf{x}$ is a design matrix composed of variables that have been centered (i.e., shifted to a sample mean of zero) then $(\mathbf{x}^\text{T} \mathbf{x})/(n-1)$ is the sample covariance matrix for those variables (and so $(\mathbf{x}^\text{T} \mathbf{x})^{-1}/(n-1)$ is the inverse sample covariance matrix).

Irrespective of all this, obviously the matrix $(\mathbf{x}^\text{T} \mathbf{x})^{-1}$ could never give information that is not in $\mathbf{x}$, since it is a function of this matrix (i.e., there exists a function $\mathbf{x} \mapsto (\mathbf{x}^\text{T} \mathbf{x})^{-1}$).

Ben
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