And because arithmetic science and geometric science are connected, and support one another, the full knowledge of numbers cannot be presented without encountering some geometry, or without seeing that operating in this way on numbers is close to geometry; the method is full of many proofs and demonstrations that are made with geometric figures.
Fibonacci, preface to Liber Abaci, 1202
Introduction
Episode 1 – Makélélé and Linear Algebra
Episode 2 – Methodology, Handwaving and Diagrams
Adding and Copying
Episode 3 – Adding (Part 1) and Mr Fibonacci
Episode 4 – Dumbing Down and Magic Lego
Episode 5 – Spoilers, Adding (Part 2) and Zero
Episode 6 – Crema di Mascarpone and Diagrammatic Reasoning
Episode 7 – Copying, Discarding and The Slogan
Episode 8 – When Adding met Copying…
Episode 9 – Natural numbers, diagrammatically
Matrices and PROPs
Episode 10 – Paths and Matrices
Episode 11 – From Diagrams to Matrices
Episode 12 – Monoidal Categories and PROPs (Part 1)
Episode 13 – PROPs (Part 2) and Permutations
Episode 14 – Homomorphisms of PROPs
Episode 15 – Matrices, diagrammatically
Episode 16 – Trust the Homomorphism, for it is Fully Faithful
Integers and Relations
Episode 17 – Maths with Diagrams
Episode 18 – Introducing the Antipode
Episode 19 – Integer matrices
Episode 20 – Causality, Feedback and Relations
Episode 21 – Functions and Relations, diagrammatically
Episode 22 – The Frobenius Equation
Episode 23 – Frobenius Snakes and Spiders
Fractions and Spaces
Episode 24 – Bringing it all together
Episode 25 – Fractions, diagrammatically
What software do you use to create diagrams?
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Omnigraffle
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As an electrical engineer, I’ve long enjoyed graphical linear algebra by way of schematics. I’m also working on a graphical algebra for computations. This blog is illuminating and inspiring – thanks for sharing!
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Beautifully written, LOVED IT! And your way of presenting them as episodes really grew on me, I feel as though it’s more coherent that way.
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Any chance of there being an RSS feed of this? Yes, I know it’s antequated, but such is my workflow :)
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Try putting the address graphicallinearalgebra.net directly in your RSS reader — it works for me!
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So it does! I completely missed that, thanks!
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Have you considered trying to figure out a diagrammatic representation for continued fractions? http://perl.plover.com/yak/cftalk/INFO/gosper.txt
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Nice suggestion :) Actually someone asked me how to do irrational numbers with graphical linear algebra — continued fractions seem really natural in this setting!
I’m just coming back from a holiday, and I will first write up how to do (ordinary) fractions. And then, yes, continued fractions will be on the agenda.
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More nice material about continued fractions, chapter 14 at http://cs.nyu.edu/~yap/book/alge/ftpSite/
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Hi Pawel, Cool blog, I should have been reading this long ago!
Have you thought about using a form of bisimulation as the equivalence on graphs? This is what I ended up doing all those years ago at http://fpl.cs.depaul.edu/ajeffrey/premon/ (the interesting bit being Appendix A of http://fpl.cs.depaul.edu/ajeffrey/premon/appendices.ps.gz). Sorry about the Java applet, what can I say, it was 1998.
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Hi Alan,
Thanks for your comments!
We have an operational semantics for signal flow graphs, and we wrote it up here: http://users.ecs.soton.ac.uk/ps/papers/popl15.pdf. Ultimately we use a kind of trace equivalence as operational equivalence. For our simple signal flow graphs, where there is one global execution clock, traces are enough. On the other hand, if we were to consider concurrency/multi-threaded executions in this kind of framework then I would expect bisimulation to play a role. It’s definitely interesting the case of Petri nets, which also has a graphical semantics; but also there we got more mileage out of traces because we were interested in using compositionality of the algebra for model checking, and for something like reachability traces are enough. Check out http://users.ecs.soton.ac.uk/ps/papers/rp2014.pdf.
I took a look at your paper; I like the way it handles both linear and non-linear components: your graphical syntax is very close to what we’ve been doing! Your operational semantics is also not very different from the labelled transition systems that come up in our work, but we typically have two labels on each transition that capture “what is observed on the dangling wires” in that time instant. It seems to me that the main difference between our approaches is that we have not really been dealing very much with traced categories (in fact, these only show up when we start insisting on directed flow); for us compact categories are the norm. For ideological reasons we are against directed flows ;)
It’s interesting that there was so much activity in this area in the late 90s. I’m trying to bring it back in, like flannel shirts ;)
All the best,
Pawel.
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I would love to be able to translate yout blog into Spanish.
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Wow, are you serious? That would be totally amazing!
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Even as a grad student in compsci I have to admit – I don’t feel like I’m intuitively comfortable with the idea of matrices, vector spaces and so forth. I understand what they are and how to use them, but the language of linear algebra still often feels foreign to me (why do we care about vector spaces and not some other structure with scalars and vectors?). This blog has been so far giving me what I’ve been looking for for a while. Keep it coming :).
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Hi Pavel
Fantastic blog, can’t wait for the next one!
Do you have a construction for the Reals?
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Thanks! I’m really looking forward to writing the next episode as well, but I’ve been totally overloaded with work in the last couple of weeks. Luckily it’s looking like this weekend will be relatively free, so there should be an episode next week.
As far as the reals go, kind of: it will follow the same idea as the continued fraction representation, but I have not written down all the details yet.
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