Interactive Linear Algebra (2019)

I think the trouble with LA education is that LA is just so ubiquitous and can look very different dependent on context. No style is going to suit everyones needs.

I wrote an entry on Wikipedia on visualizing the QR algorithm: https://en.wikipedia.org/wiki/QR_algorithm

The visualization helped me spot an unstable fixed point and understand the behaviour of the algorithm near eigenvalue clashes. The behaviour's quite sophisticated.

I think I wrote this there, but here goes again. The idea is that a positive-definite symmetric matrix can be visualized as an ellipse. This follows from the spectral theorem. Each iteration of the QR algorithm causes the ellipse to fall towards the x-axis, as if under the influence of gravity. The unstable fixed point corresponds to when the ellipse is standing up precariously, unable to fall in either direction. If you tilt it by just a bit, it will fall over (so the fixed point is unstable).

The case when the ellipse is nearly circular (corresponding to near eigenvalue clashes) causes the ellipse to fall over slowly. I think this also makes physical sense, if you think of it being under the influence of gravity. If you think of this near-circle as being a matrix, then this matrix is nearly equal to a scalar multiple of the identity matrix, so its eigenvalues are essentially known. The fact that the ellipse falls very slowly implies that the eigenvectors are unstable near eigenvalue clashes, but the eigenvalues are easy to find.

Note: The issues surrounding the unstable fixed point can be fixed using Wilkinson shifts. This makes each iteration into a discontinuous function, allowing all the fixed points to be stable. The issue surrounding instability of the eigenvectors near eigenvalue clashes cannot be fixed, as it's intrinsic to eigendecomposition (even of symmetric matrices). The latter difficulties can be dodged by slightly perturbing the matrix, but the resulting eigenvectors can be very different from the eigenvectors of the unperturbed matrix.

> an example where print publishing does a disservice

What would be an example (in the area of learning) where it does not? In the age of pervasive computing printed text is just that - a printout. A screenshot, if you will. Not extremely useful. Even a basic 3D model of something (like a car engine) that you could simply rotate and zoom is immensely more informative than a static picture of the same object.

Text, such as a mathematical proof, too, could be made dynamic for the benefit of the student: you could hide or show references, insert notes, etc.

Static/printed content is merely an artifact of the past technology we continue clinging to for no good reason.

To say nothing about the environmental impact.

Since you mentioned QR, I want to link [1] to one of my favorite articles, which explains how to use the QR decomposition to sample a random orthogonal matrix uniformly at random. The basic idea, of course, is that if A is a random matrix, and we compute A = QR, then Q is a random orthogonal matrix. But with what distribution? Well, it turns out to be almost uniform (according to Haar measure), but there are some density issues. A simple scaling of QR decomposition resolves the issue.

When I was a TA I loved giving this as an example to students. It's a great teaser for the rabbit hole that is random matrix theory.

[1] Mezzadri 2002, "How to Generate Random Matrices from the Classical Compact Groups" (http://www.ams.org/notices/200705/fea-mezzadri-web.pdf)

> I think that textbooks, math textbooks in particular, are an example where print publishing does a disservice. (I'm counting PDFs here too.) By having to lay everything out in print form, you have to clutter up your explanations with examples and footnotes that take up physical room. .

I agree with your sentiment. Although, I believe there is bit of problem of generalizing it to a wider field. Surely, you can make explorables with Linear Algebra, Trigonometry, Geometry, Number Theory etc, but there still exists a lot of mathematical fields, where such explanation using visualization is pretty difficult. It takes a lot of time to come up good explanations, while even longer for interactive explorables.

> All in all, I'd love to see more interactive textbooks. We've got this really expressive kind of document via the web. I think we should be taking advantage of it more.

This is something I have spent the last 4 years of my life building. I designed an alternative way of writing interactive books that takes a different approach than using just explorables. It's a conversational way of learning subjects designed for auto-didacts.

You interact (converse?) with the book and learn gradually. You can of course embed explorables into it (which I am planning to for upcoming linear algebra / discrete mathematics course), but the general process can still be generalized to any field. And if you are a LaTeX fan, you are in luck as all your notes are compiled as a Tufte-Latex Notebook. The medium is called Primer, homage to the diamond age.

You can checkout Primer here https://primerlabs.io.

A comic-based guide to understand how Primer helps you learn effectively: https://primerlabs.io/comics/introducing-primer-comics/