# Bilinear maps and Bilinear algorithms

1. How can one intuitively understand the definition of a bilinear map? Is there some way of looking at it geometrically? I found the following definition:

Let $\mathit{A}$,$\mathit{B}$,$\mathit{C}$ be vector spaces. A map $f:\mathit{A}\times \mathit{B}\to C$ is said to be bilinear if for each fixed element $b\in \mathit{B}$, $f(.,b):\mathit{A}\to\mathit{C}$ is a linear map. Similarly, for each fixed element of $\mathit{A}$.

Matrix multiplication is an example of a bilinear map.Following my definition, I can prove that it is a bilinear map, but I don't understand the intuitive idea behind it. In my opinion, it is simply a linear map with one element fixed.

1. Is there some formal definition of a bilinear algorithm? I could find an explanation for it only in the context of matrix multiplication: http://www.issac-symposium.org/2014/tutorials/ISSAC2014_tutorial_handout_LeGall.pdf

Kindly help me with these questions. Thanks!

• I don't think "bilinear algorithm" is a general term. In the paper you link to it appears to be the author's own description of a class of algorithm for matrix multiplication in particular. I've never heard "bilinear" used as a property of algorithms in general. Conceivably the word could be used about algorithms with two distinct inputs and $O(mn)$ time complexity, but that's not exactly a standard usage. – Henning Makholm Aug 19 '15 at 17:25
• an important example urgent to be settled is to study bilinear maps $V\times V\to{\Bbb F}$ where $V$ is a finite dimensional vectorspace over a field $\Bbb F$ – janmarqz Aug 19 '15 at 17:28
• @HenningMakholm How about my first question? – Neha Aug 19 '15 at 18:24

I think there are many explanations of what it is, but as far as intuition is concerned I think the following is by far the most satisfying. We see bilinear maps as generalizations of the properties of a product, for instance if $K$ is a field or even a ring then the map $\times: K \times K \to K$ is a bilinear map, and this is the starting point, in a sense, of a very robust class of bilinear maps that occur in mathematics. The rules that define the tensor product (the unique vector space where its linear maps are the same as bilinear maps on the corresponding product) are all just simple properties of this pairing.

Edit: I was trying to point out that a motivating example for bilinear maps is to look at the multiplication map $\times$, given any ring, the distributive property means that the map $(a, b) \to ab$ gives a bilinear map from $R \times R \to R$. This has always struck me as the symbological inspiration for writing a tensor product of two modules as $V \otimes W$ to show the relation to the product. Further all of the axiomatic properties that describe $V \otimes W$ as a quotient of $V \times W$ can be interpreted as the relation that a product in a non-commutative ring would satisfy, complete with some properties of multiplication by scalars to make the tensor product an $R$-module.

This makes it clear that the tensor product of vector spaces or modules is intended to be some kind of "vector-space generalization" of the multiplication of numbers, which is made clear by its dimension formula. By the same token the direct sum is supposed to be a generalization of addition in some sense.

• I'm not sure if I understand your answer. Could you please elaborate a bit more? – Neha Aug 19 '15 at 22:25
• I made an edit. Let me know if you're still confused about what I'm saying. – Sempliner Aug 20 '15 at 0:33
• developing a little more the idea, maps like $V\times V\to\Bbb R$ allows to consider metric properties (lengths, angles, area, volume, etc.. ) in $V$ – janmarqz Aug 20 '15 at 19:17

For a definition of a bilinear algorithm, please look at:

Michael Kaminski, David G Kirkpatrick, Nader H Bshouty, Addition requirements for matrix and transposed matrix products, Journal of Algorithms, Volume 9, Issue 3, 1988, Pages 354-364.