Tensor product of quadratic forms information

In mathematics, the tensor product of quadratic forms is most easily understood when one views the quadratic forms as quadratic spaces. If R is a commutative...

v\otimes w} is called the tensor product of v and w. An element of V ⊗ W {\displaystyle V\otimes W} is a tensor, and the tensor product of two vectors is sometimes...

tensor, curvature tensor, ...), and others. In applications, it is common to study situations in which a different tensor can occur at each point of an...

correspondence between quadratic forms and symmetric bilinear forms breaks down. By the universal property of the tensor product, there is a canonical...

A metric tensor g is positive-definite if g(v, v) > 0 for every nonzero vector v. A manifold equipped with a positive-definite metric tensor is known...

differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional...

\otimes } is the outer product. Let R {\displaystyle \mathbf {R} } be the matrix that represents a body's rotation. The inertia tensor of the rotated body is...

from the tensor algebra. See the article on tensor algebras for a detailed treatment of the topic. The exterior product of multilinear forms defined above...

vector space with a quadratic form. The Clifford algebra of U + V is isomorphic to the tensor product of the Clifford algebras of U and (−1)dim(U)/2dV...

F of characteristic not 2. For a natural number n, an n-fold Pfister form over F is a quadratic form of dimension 2n that can be written as a tensor product...

antisymmetric covariant tensor field of rank k. The differential forms on M are in one-to-one correspondence with such tensor fields. As well as the addition...

topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products (see...

The Witt group of k can be given a commutative ring structure, by using the tensor product of quadratic forms to define the ring product. This is sometimes...

graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which...

(alternatively, wedge product). The inner product is more correctly called a scalar product in this context, as the nondegenerate quadratic form in question need...

of M-theory. In the mid-1980s, a decade after Lovelock proposed his generalization of the Einstein tensor, physicists began to discuss the quadratic Gauss–Bonnet...

coordinates. The Minkowski inner product is not an inner product, since it is not positive-definite, i.e. the quadratic form η(v, v) need not be positive...

the signature (v, p, r) of a metric tensor g (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional...

not form biquaternion algebras in this sense. Let F be a field of characteristic not equal to 2. A biquaternion algebra over F is a tensor product of two...

on some basis, the bilinear form is the dot product, or, equivalently, the quadratic form is the sum of the square of the coordinates. All orthogonal...

the notion of ε-quadratic forms and ε-symmetric forms. In terms of representation theory: exchanging variables gives a representation of the symmetric...

to their exterior product. Most instances of geometric algebras of interest have a nondegenerate quadratic form. If the quadratic form is fully degenerate...

mathematics, a Killing tensor or Killing tensor field is a generalization of a Killing vector, for symmetric tensor fields instead of just vector fields....

into the tensor algebra; this is used to give the tensor algebra a basis as well. This is done by lifting: given some arbitrary sequence of e a {\displaystyle...