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For OLS in matrix form, we are taught that Hat matrix is $X(X^TX)^-X^T$, and is idempotent etc, i.e. when it multiplies with itself, it will self cancel and thus lead back to the same Hat matrix.

I was wondering why can't the inverse in the middle just unravel itself, thus causing two self cancellations (due to X multiplying with its inverse, likewise for $X^T$), but this self-implosion can only occur when we mutliply the Hat onto something else (like when we multiply it with itself, for e.g.)

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    $\begingroup$ Could you tell us what you mean by the "inverse" of a non-square matrix $X$? $\endgroup$
    – whuber
    Commented Dec 10, 2020 at 16:11
  • $\begingroup$ Ah... I see now. Thank you so much, clearly need to brush up my linear algebra... $\endgroup$
    – jojorabbit
    Commented Dec 11, 2020 at 7:27

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As you observed, the inverse is a distributive operation, thus in principle it could be applied separately to terms in a product.

However, as whuber mentioned in comments, the actual inverse operation is not defined for non-square matrices.

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