Multilinear algebra

Multilinear algebra is the study of functions with multiple vector-valued arguments, which are linear maps with respect to each argument. Concepts such as matrices, vectors, systems of linear equations, higher-dimensional spaces, determinants, inner and outer products, and dual spaces emerge naturally in the mathematics of multilinear functions. Multilinear algebra is a foundational mathematical tool in engineering, machine learning, physics, and mathematics.
Origin
While many theoretical concepts and applications are concerned with single vectors, mathematicians such as Hermann Grassmann considered the structures involving pairs, triplets, and general multi-vectors that generalize vectors. With multiple combinational possibilities, the space of multi-vectors expands to 2n dimensions, where n is the dimension of the relevant vector space. The determinant can be formulated abstractly using the structures of multilinear algebra. Multilinear algebra appears in the study of the mechanical response of materials to stress and strain, involving various moduli of elasticity. The term "tensor" describes elements within the multilinear space due to its added structure. Despite Grassmann's early work in 1844 with his Ausdehnungslehre, which was also republished in 1862, the subject was initially not widely understood, as even ordinary linear algebra posed many challenges at the time.
The concepts of multilinear algebra find applications in certain studies of multivariate calculus and manifolds, particularly in relation to the Jacobian matrix. Infinitesimal differentials encountered in single-variable calculus are transformed into differential forms in multivariate calculus, and their manipulation is carried out using exterior algebra.
Following Grassmann, developments in multilinear algebra were made by Victor Schlegel in 1872 with the publication of the first part of his System der Raumlehre and by Elwin Bruno Christoffel. Notably, significant advancements came through the work of Gregorio Ricci-Curbastro and Tullio Levi-Civita, particularly in the form of absolute differential calculus within multilinear algebra. Marcel Grossmann and Michele Besso introduced this form to Albert Einstein, and in 1915, Einstein's publication on general relativity, explaining the precession of Mercury's perihelion, established multilinear algebra and tensors as important mathematical tools in physics.
In 1958, Nicolas Bourbaki included a chapter on multilinear algebra titled "Algèbre Multilinéaire" in his series Éléments de mathématique, specifically within the book on algebra. The chapter covers topics such as bilinear functions, the tensor product of two modules, and the properties of tensor products.
Topics in multilinear algebra
The field of multilinear algebra has experienced many changes in its presentation over the years. The following pages provide additional information that is central to the topic:
Multivector
Geometric algebra
Clifford algebra
Closed and exact differential forms
Component-free treatment of tensors
Cramer's rule
Dual space
Einstein notation
Exterior algebra
Inner product
Outer product
Kronecker delta
Levi-Civita symbol
Multilinear form
Pseudoscalar
Pseudovector
Spinor
Tensor
Tensor algebra, Free algebra
Tensor contraction
Symmetric algebra, Symmetric power
Symmetric tensor
Mixed tensor
There is also a glossary available for tensor theory.
Applications
Multilinear algebra concepts find applications in various areas, including:
Classical treatment of tensors
Dyadic tensor
Metric tensor
Bra–ket notation
Hypercomplex number
Multilinear subspace learning
References
Greub, W. H. (1967) Multilinear Algebra, Springer
Douglas Northcott (1984) Multilinear Algebra, Cambridge University Press