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I came across this formula as part of a proof in linear regression and would really appreciate if someone can provide insight.

$\Sigma_{i,j}(I-H)(i,j)\epsilon_i\epsilon_j$

where $I,H$ are matrices and $\epsilon$ is a vector.

I am confused about the portion $(i,j)\epsilon_i\epsilon_j$ in the above formula. What does it mean and how do I expand the terms?

Thanks for any pointers!

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  • $\begingroup$ Perhaps $I$ and $H$ are not vectors but matrices? Perhaps $(I-H)(i,j)$ means the element at the $i$-th row, $j$-th column of the matrix $I-H$? I don't know. If the document you are reading does not establish the notation, that's the author's fault. $\endgroup$ – norio Mar 29 at 19:41
  • $\begingroup$ You are correct. I had a typo. Indeed I and H are matrices. I will edit my question. $\endgroup$ – biostat Mar 29 at 19:47
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For notation (assuming as you say, $I,H$ are matrices and $\epsilon$ is a vector):

$\bullet$ $(I-H)(i,j)$ should be the $i^{th}$ row, $j^{th}$ column of the matrix resulting from the subtraction of matrix $H$ from matrix $I$

$\bullet$ $\epsilon_i$ should be the $i^{th}$ entry of the vector $\epsilon$

$\bullet$ $\epsilon_j$ should be the $j^{th}$ entry of the vector $\epsilon$

And since the sum is over $i$,$j$, you can sum over all values of $i$ first, then each value of $j$, so $$ \sum_{j}\left(\sum_{i} (I-H)_{(i,j)}\epsilon_{i}\epsilon_{j} \right) $$

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  • $\begingroup$ thanks. i was only confused because (i,j) was not typed as a subscript to (I-H). Could it be a typo? $\endgroup$ – biostat Mar 29 at 20:35
  • $\begingroup$ it could be a typo, or maybe the author adopts that notation. I feel like most authors would write it as a subscript $\endgroup$ – NazimJ Mar 29 at 20:37

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