1. Technically, the table you shared is better thought of as a two-dimensional tensor, rather than a "graph-like matrix" -- which as you point out must be a linear map from a (vector) space to itself.
2. While not technically "Principal Component Analysis", one could do "Singular Value Decomposition" for an arbitrarily shaped 2-tensor. Further, there are other decomposition schemes that make sense for more generic tensors.
3. (Rotations / linear combinations in such spaces) Given a table of stock holdings, it can be sensible to talk about linear combinations / rotations etc. Eg: The "singular vectors" in this space could give you a decomposition in terms of companies held simultaneously by people (eg: SAAS, energy sector, semiconductors, entertainment, etc). Likewise, singular vectors on the other side would tell you the typical holding patterns among people (and clustering people by those, eg. retired pensioner invested for steady income stream, young professional investing for long-term capital growth, etc).
As it turns out, this kind of approximate (low-rank) factorization is at the heart of recommender systems.
1. Technically, the table you shared is better thought of as a two-dimensional tensor, rather than a "graph-like matrix" -- which as you point out must be a linear map from a (vector) space to itself.
2. While not technically "Principal Component Analysis", one could do "Singular Value Decomposition" for an arbitrarily shaped 2-tensor. Further, there are other decomposition schemes that make sense for more generic tensors.
3. (Rotations / linear combinations in such spaces) Given a table of stock holdings, it can be sensible to talk about linear combinations / rotations etc. Eg: The "singular vectors" in this space could give you a decomposition in terms of companies held simultaneously by people (eg: SAAS, energy sector, semiconductors, entertainment, etc). Likewise, singular vectors on the other side would tell you the typical holding patterns among people (and clustering people by those, eg. retired pensioner invested for steady income stream, young professional investing for long-term capital growth, etc). As it turns out, this kind of approximate (low-rank) factorization is at the heart of recommender systems.