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Which is the first option in the tensor product?
The first option gives a new list of n +m n + m numbers, while the second option gives a new list of nm n m numbers. The first gives a way to build a new space where the dimensions add; the second gives a way to build a new space where the dimensions multiply.
How is the tensor product computed for vector spaces?
Computing the tensor product. For vector spaces, the tensor product V ⊗ W is quickly computed since bases of V of W immediately determine a basis of V ⊗ W, as was mentioned above. For modules over a general (commutative) ring, not every module is free. For example, Z/nZ is not a free abelian group (= Z-module).
How is the tensor product similar to the Cartesian product?
Forming the tensor product v⊗w v ⊗ w of two vectors is a lot like forming the Cartesian product of two sets X×Y X × Y. In fact, that’s exactly what we’re doing if we think of X X as the set whose elements are the entries of v v and similarly for Y Y .
Is the universal property of a tensor product valid?
The universal-property definition of a tensor product is valid in more categories than just the category of vector spaces. Instead of using multilinear (bilinear) maps, the general tensor product definition uses multimorphisms.
What is the density matrix on the tensor product?
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.
What do you call an array of tensors?
These arrays are called tensors and whenever you do a bunch of these processes together, the resulting mega-process gives rise to a tensor network. But manipulating high-dimensional arrays of numbers can get very messy very quickly: there are lots of numbers that all have to be multiplied together.
What is the order, degree or rank of a tensor?
The total number of indices required to identify each component uniquely is equal to the dimension of the array, and is called the order, degree or rank of the tensor. However, the term “rank” generally has another meaning in the context of matrices and tensors.
Is the stress tensor a second order tensor?
Since the stress tensor describes a mapping that takes one vector as input, and gives one vector as output, it is a second-order tensor. In mathematics, a tensor is an algebraic object that describes a (multilinear) relationship between sets of algebraic objects related to a vector space.
Can a tensor product be defined over a commutative ring?
More generally, the tensor product can be defined in the same way for modules over a commutative ring and abelian groups (that are modules over the integers). For vector spaces and modules that have additional structures, the tensor product is often equipped with a similar structure.