1.

Equivalently,

In the language of Category theory, the mixed-product property of the Kronecker product (and more general tensor product) shows that the category MatF of matrices over a field F, is in fact a monoidal category, with objects natural numbers n, morphisms n → m are n -by- m matrices with entries in F, composition is given by matrix multiplication, identity arrows are simply n × n identity matrices In, and … Thenthereis alinearmapφb: M⊗ N→ P suchthatφ(m,n) = φb(m⊗ n).
They may be thought of as the simplest way to combine modules in a meaningful fashion.

If v belongs to V and w belongs to W, then the equivalence class of (v, w) is denoted by v ⊗ w, which is called the tensor product of v with w. In physics and engineering, this use of the "⊗" symbol refers specifically to the outer product operation; the result of the outer product v ⊗ w is one of the standard ways of representing the equivalence class v ⊗ w. An element of V ⊗ W that can be written in the form v ⊗ w is called a pure or simple tensor. The diagram for universal property can be seen in gure 1 below. : V W!V W which has the universal property.

The number of simple tensors required to express an element of a tensor product is called the tensor rank (not to be confused with tensor order, which is the number of spaces one has taken the product of, in this case 2; in notation, the number of indices), and for linear operators or matrices, thought of as (1, 1) tensors (elements of the space V ⊗ V ), it agrees with matrix rank. Existence of the Universal Property: The tensor product has what is called a universalproperty. Definition and properties of tensor products The DFT, the DCT, and the wavelet transform were all defined as changes of basis for vectors or functions of one variable and therefore cannot be directly applied to higher dimensional data like images. On the right hand side, the symmetric product (denoted by the symbol ⊙ or by juxtaposition) has been taken. In this chapter we will introduce asimplerecipeforextendingsuchone-dimensionalschemestotwo(andhigher) (associativity) For a right R1-module M01, an R1-R2-bimodule M12, and a left R2-mo…

the type that expects two contravariant vectors as arguments. Lemma 3.1 Supposethatφ: M×N→ P isabilinearmap.

Elements of V ⊗ W are often referred to as tensors, although this term refers to many other related concepts as well.

In general, an element of the tensor product space is not a pure tensor, but rather a finite linear combination of pure tensors.

As we will see, polynomial rings are combined as one might hope, so that R[x]


The tensor product of V and W denoted by V W is a vector space with a bilinear map. For example, if v1 and v2 are linearly independent, and w1 and w2 are also linearly independent, then v1 ⊗ w1 + v2 ⊗ w2 cannot be written as a pure tensor. The tensor product (denoted by the symbol ⊗) yields a tensor field of type (0, 2), i.e. The state of that two-particle system can be described by something called a density matrix ρ ρ on the tensor product of their respective spaces Cn ⊗Cn C n ⊗ C n. A density matrix is a generalization of a unit vector—it accounts for interactions between the two particles.

The Tensor Product Tensor products provide a most \natural" method of combining two modules. the name comes from the fact that the construc-tion to follow works for all maps of the given type.

In otherwords, if ˝ : V W !Z, then there exists a unique linear map, up to isomorphism, ˝~ : V W)Zsuch that ~˝ = ˝.