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The tensor product of two vector spaces is a vector space that is defined up to an isomorphism.There are several equivalent ways to define it. Most consist of defining explicitly a vector space that is called a tensor product, and, generally, the equivalence proof results almost immediately from the basic properties of the vector spaces that are so defined.
Mathematically vectors are elements of a vector space over a field, and for use in physics is usually defined with = or .Concretely, if the dimension = of is finite, then, after making a choice of basis, we can view such vector spaces as or .
Programming languages that implement matrices may have easy means for vectorization. In Matlab/GNU Octave a matrix A can be vectorized by A(:). GNU Octave also allows vectorization and half-vectorization with vec(A) and vech(A) respectively. Julia has the vec(A) function as well.
The tensor product of V and its dual space is isomorphic to the space of linear maps from V to V: a dyadic tensor vf is simply the linear map sending any w in V to f(w)v. When V is Euclidean n-space, we can use the inner product to identify the dual space with V itself, making a dyadic tensor an elementary tensor product of two vectors in ...
An elementary example of a mapping describable as a tensor is the dot product, which maps two vectors to a scalar. A more complex example is the Cauchy stress tensor T, which takes a directional unit vector v as input and maps it to the stress vector T (v), which is the force (per unit area) exerted by material on the negative side of the plane ...
The tensor can be flattened in three ways to obtain matrices comprising its mode-0, mode-1, and mode-2 vectors. [ 1 ] In multilinear algebra , mode-m flattening [ 1 ] [ 2 ] [ 3 ] , also known as matrixizing , matricizing , or unfolding , [ 4 ] is an operation that reshapes a multi-way array A {\displaystyle {\mathcal {A}}} into a matrix denoted ...
A dyadic tensor T is an order-2 tensor formed by the tensor product ⊗ of two Cartesian vectors a and b, written T = a ⊗ b.Analogous to vectors, it can be written as a linear combination of the tensor basis e x ⊗ e x ≡ e xx, e x ⊗ e y ≡ e xy, ..., e z ⊗ e z ≡ e zz (the right-hand side of each identity is only an abbreviation, nothing more):
In mathematics, the signature (v, p, r) [clarification needed] of a metric tensor g (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and zero eigenvalues of the real symmetric matrix g ab of the metric tensor with respect to a basis.