Let me start by saying that I am starting to fall in love with the spherical harmonics and analysis of functions defined on a sphere. I am a physicist studying Cosmology, so you can imagine I get to work with them quite often. However, more often than not I have to work with incomplete sky fraction (incomplete sphere) and my question is related to that. Here is my question:
I am currently trying to read this paper. I am confused by the statements made about eq. 1 in the lines immediately prior to it describing this equation. The authors say, that this equation encodes the orthonormality properties of a set of functions. I am unable to understand how this relation is equivalent to a matrix? At the end of the previous section, the authors describe $Y^T_i$ as the set of spherical harmonic functions $Y_{lm}$ where the indices, $l$ & $m$ are grouped in some form into $i (l,m)$ to make $Y^T_i$ a vector. What I am not able to understand is this: By taking this definition, $Y(\hat{r})Y^T(\hat{r})$ is then a dot product which will result in a single expression. This expression is simply a list of $Y_i(\hat{r})^2$ which is not a matrix. This understanding is further confirmed eq. 6 of the paper where they further expand the dot product explicitly, or may be I am not understanding the mathematics at all. However, the text refers to the coupling matrix as a matrix.
Thank you for your time.