Orthogonal transformation of standard normal sample

Question

I was reading this pdf https://ocw.mit.edu/courses/mathematics/18-443-statistics-for-applications-fall-2003/lecture-notes/lec15.pdf

Shouldn't

$Var(Y_i)=\sum_{k=1}^nv_{ki}^2$ (from how $Y_i$ is defined)

instead of

$$Var(Y_i)=\sum_{k=1}^nv_{ik}^2$$ what author has written.

1. But if what author has written is correct, can you please explain why?

2. Also How $EY_iY_j=\sum_{k=1}^nv_{ik}v_{jk}$?

Thanks.

Answer 1

Yes, technically $\displaystyle{Var(Y_i) = \sum_{k=1}^n v_{ki}^2}$ would be the more appropriate equation. But actually it doesn't really matter and $\displaystyle{\sum_{k=1}^n v_{ki}^2 = \sum_{k=1}^n v_{ik}^2 = 1}$. This is because if $O$ is an orthogonal matrix, then $O^T$ is also an orthogonal matrix.

$$\mathbb{E}[Y_i Y_j] = \mathbb{E}\left[ \left(\sum_{k=1}^n v_{ki} X_i \right) \left(\sum_{k=1}^n v_{kj} X_i \right) \right]$$

If you expand that out, all the cross terms vanish because $\mathbb{E}[X_i X_j] = 0$ for $i \neq j$ (because $X_i$ and $X_j$ are uncorrelated).