Multilinear Algebra - Origin

Origin

In a vector space of dimension n, one usually considers only the vectors. According to Hermann Grassmann and others, this presumption misses the complexity of considering the structures of pairs, triples, and general multivectors. Since there are several combinatorial possibilities, the space of multivectors turns out to have 2n dimensions. The abstract formulation of the determinant is the most immediate application. Multilinear algebra also has applications in mechanical study of material response to stress and strain with various moduli of elasticity. This practical reference led to the use of the word tensor to describe the elements of the multilinear space. The extra structure in a multilinear space has led it to play an important role in various studies in higher mathematics. Though Grassmann started the subject in 1844 with his Ausdehnungslehre, and re-published in 1862, his work was slow to find acceptance as ordinary linear algebra provided sufficient challenges to comprehension.

The topic of multilinear algebra is applied in some studies of multivariate calculus and manifolds where the Jacobian matrix comes into play. The infinitesimal differentials of single variable calculus become differential forms in multivariate calculus, and their manipulation is done with exterior algebra.

After some preliminary work by Elwin Bruno Christoffel, a major advance in multilinear algebra came in the work of Gregorio Ricci-Curbastro and Tullio Levi-Civita (see references). It was the absolute differential calculus form of multilinear algebra that Marcel Grossman and Michele Besso introduced to Albert Einstein. The publication in 1915 by Einstein of a general relativity explanation for the precession of the perihelion of Mercury, established multilinear algebra and tensors as physically important mathematics.