Here at “Picture this Maths”, we were very lucky last month to be featured by the American Mathematical Society (AMS) on their Blog on Math Blogs! It is wonderful to have people reading and sharing our blog in the mathematical community and beyond.

This blog post also tells an exciting AMS story. It is on the topic of tensors (like this post, and this one, and this one too). It’s about a mathematical picture which started out as a cartoon in my notebook.

It ended up — much souped up, with help from computer graphics experts — on the front cover of the June/July issue of the Notices of the AMS. The full issue is available here.

So, what’s going on in this picture?

The story begins, as with many stories (ok, many of my stories) with singular vectors and singular values of matrices. To understand mathematical concepts, it’s useful to have a picture in mind. Luckily, singular vectors and singular values of matrices lend themselves extremely well to visual description. Just take a look at this wikipedia gif.

A matrix can be thought of in complementary ways, either as a two-dimensional grid of data, or as the information that encodes a linear transformation of a space. The gif is about matrices as linear maps. Below are a couple of still images from it. They show how a linear transformation of space

can be decomposed as the combination of three “building block” transformations, each of which is far easier to understand. A rotation , a coordinate scaling and then another rotation

What about visualizing the singular vectors and singular values of tensors?

Here, the story is more complicated, not least because the greater number of dimensions makes visualizing things harder. Usually, matrices have a finite number of singular vectors, and the same is true of tensors. But, like for the matrix case, some tensors have infinitely many singular vectors, and the singular vectors themselves form an interesting structure.

The picture shows the structure of the singular vectors of a four-dimensional orthogonally decomposable tensor of size . For more on the ‘maths behind the picture’, see this About The Cover article from the AMS.

Do you mean that a tensor has finitely many singular values, but it can have infinitely many singular vectors? Right now both sentences say singular vectors.

I am only talking about singular vectors, singular vectors of matrices and singular vectors of tensors. Just as some (well, most) matrices have finitely many singular vectors:
-for example, the singular vectors of the 2×2 matrix (2 0; 0 1) are (1; 0) and (0; 1)
there are other matrices that have infinitely many singular vectors
-for example, the identity matrix of size 2 has “all vectors of length two” as its singular vectors.

This same non-generic behaviour can occur for tensors too. While most tensors have a finite list of singular vectors, there are other tensors that have infinitely many.

kaieAugust 29, 2016 / 10:42 amDo you mean that a tensor has finitely many singular values, but it can have infinitely many singular vectors? Right now both sentences say singular vectors.

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annaAugust 29, 2016 / 5:31 pmhey Kaie, thanks for your comment.

I am only talking about singular vectors, singular vectors of matrices and singular vectors of tensors. Just as some (well, most) matrices have finitely many singular vectors:

-for example, the singular vectors of the 2×2 matrix (2 0; 0 1) are (1; 0) and (0; 1)

there are other matrices that have infinitely many singular vectors

-for example, the identity matrix of size 2 has “all vectors of length two” as its singular vectors.

This same non-generic behaviour can occur for tensors too. While most tensors have a finite list of singular vectors, there are other tensors that have infinitely many.

I hope this answers your question.

Anna

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