Origin of Multi-linear map

Question

Multi-linear map is used to define the determinant. However, since the multi-linear map itself does not have linearity, I feel a sense of heterogeneity when I compare this concept with the contents of general linear algebra. So, I am curious as to why the multi-linear map was given such a name with the word linearity in it, and I am curious about the historical background and motivation at the time when the multi-linear map was created.

Answer 1

A pop pseudo-definition of multilinearity: A map is multi-linear if it is linear in each of its "components".

It's a very complicated story so this is -- despite its length -- a sketch at best.

Muir's 5 volumes on the historical development of the theory of determinants (up until 1920) are a good source to understand the determinant side of the developments.

Muir credits Leibniz in a letter to de L'Hospital to essentially do the ground work for formulating notions of the determinant. The key is that one looks to solve a set of linear equation and find numbers that characterize the equations and their solutions. With a big post-hoc lens, it may be unsurprising that in a situation with multiple linear equations we will deal with multi-linearity.

I would attribute multi-linearity in the modern sense to Hermann Grassmann (1844), and there is a nice geometric way to thing of multi-linearity. Take a rectangle. You can linearly deform each side of the rectangle. If you study the classical area formula A=x*y, it is not linear; if you manipulate both components (it's quadratic, incidentally etymologically a word for a square rectangle!), but it is linear with respect to each side, keeping the other side constant.

Grassmann's father Justus Grassmann discovered that if you account for sign, the area formula for the rectangle remains valid under linear transformation in each side, and Hermann Grassmann discovered that if you keep the sign and just follow the rules of linearity, you recover that indeed it also holds for parallelograms (i.e. the right angle is not a necessary condition for multi-linearity) and that in fact this remains valid in arbitrary dimensions.

Grassmann explains this in the introduction of his 1844 book:

• Grassmann, H. (1844) Die lineale Ausdehnungslehre - ein neuer Zweig der Mathematik, Wiegand. (compare also the English translation by Kannenberg).

He also discovers many modern notions of linear and multi-linear algebra including linear n-forms, the Hodge dual, the ability to define the inner product from the Hodge dual, and most importantly the exterior product, which would encapsulate the determinant in a basis-free way.

Grassmann is very much concerned with developing new products (around the same time Hamilton was working on a new product that would be the product of Quaternions). So this is one of the frames of the discussion. The second is geometry. Combining the two and seeking abstraction, Grassmann ends up developing in many ways what we consider modern multi-linear algebra over the reals.

In his first book of 1844 he makes no mention of determinants. Those appear in his second book (1862), intended as a more accessible/improved exposition.

Grassmann himself does not use the term multilinear, though he clearly understood the concept. I do not know with any certainty who first coined the term multi-linear. The earliest reference known to me is

• de Morgan, A. (1866). On the Conic Octagram. Proceedings of the London Mathematical Society, 1(1), 26-29.

Jacobi is sometimes credited for having coined the term bilinear form, for example in

• Kronecker, L. (1903). Vorlesungen über die Theorie der Determinanten, Erster Band, Bearbeitet und fortgeführt von K. Hensch, BG Teubner, Leipzig.

But checking the actual article by Jacobi

• Jacobi, C. G. J. (1857). Über eine elementare Transformation eines in Bezug auf jedes von zwei Variablen-Systemen linearen und homogenen Ausdrucks.Journal für die reine und angewandte Mathematik, 53, 265-270.

one finds that Jacobi actually used the "quadratic form" terminology. In Space, Time, Matter (1922) p.26 Weyl writes "bilinear and quadratic forms" juxtaposing the two notions.

The story is historically very much complicated because multiple lines of inquiry happened in parallel often with unawareness of each others development. In parallel to Grassmann's work, there was work on determinants (see Jacobi's as an example), with apparently no interface, in at least one direction. Grassmann in 1862 does refer to Jacobi's functional determinants for example.

I have the suspicion that nomenclature may only have become fully established and understood with Bourbaki:

• Bourbaki, N. (1948) Elements de Mathematique, Vol. VII., Algebre, Livre II, Algebre Multilineaire, Ch. 3. Vol. VII. Hermann.

For comparison neither van der Waerden (Algebra I,II) nor E. Artin (Geometric Algebra) uses the word multilinear, though both use bilinear.

The way all this came together is complicated, and in some sense one might think of it as still ongoing. Grassmann's work was known and fruitfully used to develop the notion of a differential form by the French school (E. Cartan, Goursat, Poincare); it was also in different ways studied by Burali-Forti and Peano in Italy and leading to Peano's form of linear algebra. Somewhat paradoxically, Grassmann's recognition in Germany was particularly poor, especially by algebraists. Klein mentions his work in his Erlangen program and would essentially form one of volumes of Elementary Mathematics from a Higher Viewpoint lectures to expose Grassmann's work in based determinant notation, but aside from Mobius and Kummer there was little contemporary resonance or awareness in Germany. Hamilton knew of it as did Gibbs when he formulated vector calculus.

In numerous readings of Grassmann the crucial understanding of multi-linearity remains obfuscated. Gibbs-Heaviside vector calculus tries to make geometric vectors to be the core type to manipulate, making everything look "linear" when it sometimes really is multi-linear. The cross-product is the easiest example of this confusion. And attempts at band-aid fixes lead to definitions such as pseudo-vectors.

In short the history of multi-linear algebra is a good example how ideas do not travel in linear ways or along a single path and that a discovery to one can be misunderstood by another, and that a set of ideas can independently, or loosely connectedly, develop to then percolate into a joint understanding.

P.S. I have left out all of tensor algebra development outside Grassmann, just one indication of the incompleteness of this sketch.