Moore-Penrose pseudoinverse and Linear relations I recently came across this website called Graphical Linear Algebra. I feel like there's a lot of insight there, but it's too monolithic for me to be able to extract it by skimming.
Episode 27 is about linear relations, a concept that I'd never heard of until reading about it there.
They discuss the concept of the converse of a linear relation (same as regular converse) and it occurred to me that it's a generalisation of the Moore-Penrose pseudoinverse (which doesn't feature in the article).
Do you know a good introduction to this topic with illuminating examples tying it to concepts in "regular" linear algebra? And in particular explains the connection between the Moore-Penrose pseudoinverse and the converse in detail?
 A: I'm really glad I stumbled on this post. The blog was really interesting and I learned a lot.
That being said, from the tone of the posts, it seems like it's really an active topic of research, and, if that's the case, any book would likely be 5-10 years from being written and published.
One of the episodes does say this, though:

Two weeks ago I was in Lyon for Fabio’s PhD defence. It’s now possible
  to download his thesis, which received high praise from the very
  impressive thesis committee. The thesis is currently the most thorough
  account of graphical linear algebra and its applications, so do take a
  look!

It comes with a link to the thesis: http://zanasi.com/fabio/IHthesis_FZ.pdf
Alternatively, if you're generally interested in the generalization of the Moore-Penrose pseudoinverse, you might be better off reading an abstract algebra book with heavy emphasis on category theory.
Finally, from reading up to Episode 29, it seems like the converse relationship is mostly established to reveal a new perspective on a pretty established part of linear algebra. So, linear relations can be thought of as: $\{(x,y) : Cx = Dy\}$ for some matrices $C$ and $D$. (This is called the co-span form in his notes.) Taking the converse simply yields $\{(y,x) : Cx = Dy\}$.
I think more detailed connections would likely be novel research, so if you have ideas, you should consider publishing them!
