There is a wikipedia article. There is a paper by Elisha Peterson. I tried reading these but they don't seem to click for me.

Are there books or other resources for learning how to do linear algebra with trace diagrams?

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    $\begingroup$ Perhaps a related question would be: Why does one want to do linear algebra with trace diagrams. I'd love a good answer. $\endgroup$
    – GMRA
    Nov 19, 2009 at 17:38
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    $\begingroup$ Grétar, it's essentially the same notation used in Cvitanović's book on Lie groups where it is used to give some elegant diagrammatic proofs: birdtracks.eu $\endgroup$
    – Dan Piponi
    Nov 19, 2009 at 18:39
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    $\begingroup$ I use such diagrams all the time: they do a good job of keeping track of complicated contractions. $\endgroup$ Nov 19, 2009 at 21:36
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    $\begingroup$ I hadn't heard of trace diagrams before, and I must admit I found the wikipedia article on them singularly unhelpful. $\endgroup$ Nov 19, 2009 at 22:16
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    $\begingroup$ To answer Gretar's question: When a tensor calculation is inherently a graph, you should draw it as a graph. When a linear notation is more concise, you should use that instead. You could use tensor diagrams for an associative algebra, but there is no point; you could use them in Riemannian curvature calculations, but with effort you can avoid them. For Hopf algebra calculations (see my answer below), tensor diagrams are at least as good as the only serious linear competitor, Sweedler notation. $\endgroup$ Nov 20, 2009 at 2:16

5 Answers 5


The best resource I can point a beginner to is the first few chapters of Stedman's book "Group Theory". He focuses on the specific example of 3-vector diagrams, and does a good job of including lots of sample calculations. Unfortunately, it's not available online. I have found Cvitanovic' book fascinating but tough to internalize. You might also try looking at some of the work of Jim Blinn (a compilation of his work is at http://research.microsoft.com/pubs/79791/UsingTensorDiagrams.pdf), which has a lot of examples worked out. Another text that I commonly referred to is "The classical and quantum 6j-symbols" (https://books.google.com/books?id=mg8ISMd5mO0C), although this is limited to a special case of the diagrams.

As for learning about the diagrams themselves, I think the only way to really get comfortable with it is to work out lots of examples. I filled endless chalkboards at the University of Maryland with the doodles... it's one of the fun parts of the subject. :)

The paper you mentioned is focused on the applications of diagrams to ideas in traditional linear algebra. I have not found any other source that focuses exclusively on this use of diagrams, although Cvitanovic' book (for example) mentions without proof that one of his equations corresponds to the Cayley-Hamilton Theorem. This is probably because many mathematicians do not see much use in reproving old results (particularly if one must learn new notation to do so). I personally feel that there is sufficient beauty and elegance (once the notation is understood) in diagrammatic proofs of these "old proofs" to make them interesting. I also think that a deeper understanding of diagrammatic techniques is a worthy goal in itself. Others have mentioned some of the existing applications.

The term "trace diagrams" originated in my thesis, so you won't find it in many published papers. I use it to mean the particular class of diagrams that are labeled by matrices. There are many other names. I first learned about them in the special case of "spin networks" (a special case), and Penrose has the strongest claim to historical priority, hence "Penrose tensor diagrams".

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    $\begingroup$ 1) I think that tensor diagrams is a better than trace diagrams. My own name, arrow notation, is not all that great either. I would suggest the former as a title for the Wikipedia page. 2) Not to toot my own horn too much, but I'm really happy that Blinn cites my old paper. Yes, Penrose was much earlier than Stedman or me, but the gist of his notation was slightly different. $\endgroup$ Nov 20, 2009 at 1:23
  • $\begingroup$ Yes, I agree that "tensor diagrams" would be better. There is actually a wikipedia page for that, "Penrose's tensor notation" I think. But the flavor of that wikipedia page is a lot different than the one on trace diagrams. $\endgroup$ Nov 20, 2009 at 2:02

to Greg Kuperberg: Use of such diagrams probably starts in mid 19th century (Sect. 4.9 A BRIEF HISTORY OF BIRDTRACKS). If you find "spiders" or “birdtracks” too silly, I vote for the somewhat unwinged "tensor diagrams."

So why give these diagrams a new name, and not call them “Feynman diagrams” or "tensor diagrams"? I needed a distinct name to distinguish them from the more traditional uses of diagrams. The difference is that here diagrams are not a mnemonic device, an aid in writing down a Feynman integral that is to be evaluated by other techniques. I did not call them "tensor diagrams" as that is too close to “invariant tensor operators”, the Wigner-Eckart theorem, and the 3n-j diagrams that are only a prelude to a computation. Here “birdtracks” are everything, all calculations are carried out in terms of birdtracks, from start to finish.


As it happens, I used trace diagrams in my old papers on 3-manifold invariants from Hopf algebras, arXiv:math/9201301 and arXiv:q-alg/9712047. At the time I hadn't heard of the name "trace diagrams" and I called it instead "arrow notation", and I included a review of the notation. The diagrams are very useful for understanding word relations in Hopf algebras.


I found a website by Elisha Peterson with links to 16 or so papers and preprints and two books one of which is available online see here:



I'd recommend the book Road to Reality by R. Penrose. He explains his birdtracks well.


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