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I am new to linear algebra and I have the following doubts:

  1. In weighted least square estimation of the system $Ax = b$ we minimize the weighted value of the error $e = b - Ax$ and the best $\hat{x}$ is given by $( A^T \Sigma^{-1}A )^{-1} A^T\Sigma^{-1} b$ where $\Sigma$ is the covariane matrix of the error $e$. Why is the covariance matrix $\Sigma{e}$ the best choice for the weighting matrix? Is there any derivation for it? Please refer me to its link or sum hints will also do.

  2. For the same linear system $e = b - Ax$ is $E(ee^T) = E(bb^T)$ given that error is unbiased (i.e. $E(e) = 0$)?

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