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Category# Multilinear algebra

Summary

Multilinear algebra is a branch of mathematics that expands upon the principles of linear algebra. It extends the foundational theory of vector spaces by introducing the concepts of p-vectors and multivectors using Grassmann algebras.
In a vector space of dimension n, the focus is primarily on using vectors. However, Hermann Grassmann and others emphasized the importance of considering the structures of pairs, triplets, and general multi-vectors, which offer a more comprehensive perspective. With multiple combinatorial possibilities, the space of multi-vectors expands to 2n dimensions. The abstract formulation of the determinant is one direct application of multilinear algebra. Additionally, it finds practical use in studying the mechanical response of materials to stress and strain, involving various moduli of elasticity. The term "tensor" emerged to describe elements within the multi-linear space due to its added structure. This additional structure has made multilinear algebra significant in various fields of higher mathematics. However, despite Grassmann's early work in 1844 with his Ausdehnungslehre, which was also republished in 1862, it took time for the subject to gain acceptance, as ordinary linear algebra posed enough challenges on its own.
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.

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Related publications (2)

Related concepts (1)

Higher-order singular value decomposition

In multilinear algebra, the higher-order singular value decomposition (HOSVD) of a tensor is a specific orthogonal Tucker decomposition. It may be regarded as one type of generalization of the matrix singular value decomposition. It has applications in computer vision, computer graphics, machine learning, scientific computing, and signal processing. Some aspects can be traced as far back as F. L. Hitchcock in 1928, but it was L. R. Tucker who developed for third-order tensors the general Tucker decomposition in the 1960s, further advocated by L.

Daniel Kressner, André Uschmajew

The higher-order singular values for a tensor of order d are defined as the singular values of the d different matricizations associated with the multilinear rank. When d≥3, the singular values are generally different for different matricizations but not c ...

Sabine Süsstrunk, Luciano Sbaiz, Roberto Costantini

Videos representing flames, water, smoke, etc. are often defined as dynamic textures: "textures" because they are characterized by redundant repetition of a pattern and "dynamic" because this repetition is also in time and not only in space. Dynamic textur ...