Multilinear algebra

Source: Wikipedia. Pages: 63. Chapters: Tensor product, Bilinear map, Cross product, Bivector, Exterior algebra, Paravector, Plücker coordinates, Hyperdeterminant, Einstein notation, Tensor product of modules, Pfaffian, Tensor field, Lagrange's identity, Multilinear subspace learning, Homogeneous polynomial, Complexification, Symmetric algebra, Trace diagram, Tensor algebra, Berezin integral, Multivector, Glossary of tensor theory, Multilinear map, Skew lines, Higher-order singular value decomposition, CP decomposition, Binet¿Cauchy identity, Discrete exterior calculus, Tensor product of algebras, Witt's theorem, Interior product, Definite bilinear form, Multilinear form, Grassmann¿Cayley algebra. Excerpt: In mathematics, a bivector or 2-vector is a quantity in geometric algebra or exterior algebra that generalises the idea of a vector. If a scalar is considered a zero dimensional quantity, and a vector is a one dimensional quantity, then a bivector can be thought of as two dimensional. Bivectors have applications in many areas of mathematics and physics. They are related to complex numbers in two dimensions and to both pseudovectors and quaternions in three dimensions. They can be used to generate rotations in any dimension, and are a useful tool for classifying such rotations. They also are used in physics, tying together a number of otherwise unrelated quantities. Bivectors are generated by the exterior product on vectors ¿ given two vectors a and b their exterior product is a bivector. But not all bivectors can be generated this way, and in higher dimensions a sum of exterior products is often needed. More precisely a bivector that requires only a single exterior product is simple; in two and three dimensions all bivectors are simple, but in higher dimensions this is not generally the case. The exterior product is antisymmetric, so negates the bivector, producing a rotation with the opposite sense, and is the zero bivector. Parallel plane segments with the same orientation and area corresponding to the same bivector .Geometrically, a simple bivector can be interpreted as an oriented plane segment, much as vectors can be thought of as directed line segments. Specifically for the bivector , its magnitude is the area of the parallelogram with edges a and b, its attitude that of any plane specified by a and b, and its orientation the sense of the rotation that would align a with b. It does not have a definite location or position. The bivector was first defined in 1844 by German mathematician Hermann Grassmann in exterior algebra, as the result of the exterior product. Around the same time in 1843 in Ireland William Rowan Hamilton discovered quaternions. It was not un