multilinear algebra

In mathematics, multilinear algebra extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concept of a tensor and develops the theory of 'tensor spaces'. In applications, numerous types of tensors arise. The theory tries to be comprehensive, with a corresponding range of spaces and an account of their relationships.

Historical background of the approach to multilinear algebra

The subject itself has various roots going back to the mathematics of the nineteenth century, in what was then called tensor analysis, or the "tensor calculus of tensor fields". It developed out of the use of tensors in differential geometry, general relativity, and many branches of applied mathematics. Around the middle of the 20th century the study of tensors was reformulated more abstractly. The Bourbaki group's treatise Multilinear Algebra was especially influential — in fact the term multilinear algebra was probably coined there.

One reason at the time was a new area of application, homological algebra. The development of algebraic topology during the 1940s gave additional incentive for the development of a purely algebraic treatment of the tensor product. The computation of the homology groups of the product of two spaces involves the tensor product; but only in the simplest cases, such as a torus, is it directly calculated in that fashion (see Künneth theorem). The topological phenomena were subtle enough to need better foundational concepts; technically speaking, the Tor functors had to be defined.

The material to organise was quite extensive, including also ideas going back to Hermann Grassmann, the ideas from the theory of differential forms that had led to De Rham cohomology, as well as more elementary ideas such as the wedge product that generalises the cross product.

The resulting rather severe write-up of the topic (by Bourbaki) entirely rejected one approach in vector calculus (the quaternion route, that is, in the general case, the relation with Lie groups). They instead applied a novel approach using category theory, with the Lie group approach viewed as a separate matter. Since this leads to a much cleaner treatment, there was probably no going back in purely mathematical terms. (Strictly, the universal property approach was invoked; this is somewhat more general than category theory, and the relationship between the two as alternate ways was also being clarified, at the same time.)

Indeed what was done is almost precisely to explain that tensor spaces are the constructions required to reduce multilinear problems to linear problems. This purely algebraic attack conveys no geometric intuition.

Its benefit is that by re-expressing problems in terms of multilinear algebra, there is a clear and well-defined 'best solution': the constraints the solution exerts are exactly those you need in practice. In general there is no need to invoke any ad hoc construction, geometric idea, or recourse to co-ordinate systems. In the category-theoretic jargon, everything is entirely natural.

Conclusion on the abstract approach

In principle the abstract approach can recover everything done via the traditional approach. In practice this may not seem so simple. On the other hand the notion of natural is consistent with the general covariance principle of general relativity. The latter deals with tensor fields (tensors varying from point to point on a manifold), but covariance asserts that the language of tensors is essential to the proper formulation of general relativity.

Some decades later the rather abstract view coming from category theory was tied up with the approach that had been developed in the 1930s by Hermann Weyl (in his celebrated and difficult book The Classical Groups). In a way this took the theory full circle, connecting once more the content of old and new viewpoints.

Content of multilinear algebra

The content of multilinear algebra has changed much less than the presentation, down the years. Here are further pages centrally relevant to it:

• dual space
• bilinear operator
• inner product
• multilinear map
• determinant
• Cramer's rule
• component-free treatment of tensors
• Kronecker delta
• tensor contraction
• mixed tensor
• Levi-Civita symbol
• tensor algebra, free algebra
• symmetric algebra, symmetric power
• exterior algebra
• exterior derivative
• Einstein notation
• symmetric tensor
• metric tensor

There is also a glossary of tensor theory.

From the point of view of applications

Consult these articles for some of the ways in which multilinear algebra concepts are applied, in various guises:

• classical treatment of tensors
• dyadic tensor
• bra-ket notation
• geometric algebra
• Clifford algebra
• pseudoscalar
• pseudovector
• spinor
• outer product
• hypercomplex number

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