# Help with summation notation in linear regression

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!

• 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. – norio Mar 29 at 19:41
• You are correct. I had a typo. Indeed I and H are matrices. I will edit my question. – biostat Mar 29 at 19:47

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 $$\sum_{j}\left(\sum_{i} (I-H)_{(i,j)}\epsilon_{i}\epsilon_{j} \right)$$

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