I am trying to simplify this expression where $$ (X(X^TX)^{-1}X^T)^T = X(X^TX)^{-1}X^T $$ should hold. However, I am not sure what to do with $((X^TX)^{-1})^T$ once I make the transpose notation into the bracket. Any hint/help would be really appreiciated!
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The inverse of a symmetric matrix is also symmetric. See this. – Amaan M Feb 09 '21 at 19:00
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Also this: https://math.stackexchange.com/questions/340233/transpose-of-inverse-vs-inverse-of-transpose – Physical Mathematics Feb 09 '21 at 19:00
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For any invertible matrix $A$, we have: $$(A^T)^{-1}=(A^{-1})^T\ . $$
Now, let $A=X^TX$. Then since $A^T=A$, we have $$ (A^{-1})^T=A^{-1}\ . $$