# Is tensor product commutative?

**Asked by: Alta Cole PhD**

Score: 4.5/5 (35 votes)

**No, it is not commutative**.

## Is the tensor product of vector spaces commutative?

The tensor product of two modules A and B over a commutative ring R is defined in exactly the same way as the tensor product of vector spaces over a field: **A ⊗ R B := F ( A × B ) / G**.

## Is the tensor product associative?

The binary tensor product is **associative**: (M_{1} ⊗ M_{2}) ⊗ M_{3} is naturally isomorphic to M_{1} ⊗ (M_{2} ⊗ M_{3}). The tensor product of three modules defined by the universal property of trilinear maps is isomorphic to both of these iterated tensor products.

## Is the tensor product symmetric?

For example, the tensor **product is symmetric**, meaning there is a canonical isomorphism: to. factors into a map. are inverse to one another by again using their universal properties.

## What is the product of two tensors?

If the two vectors have dimensions n and m, then their outer product is an **n × m matrix**. More generally, given two tensors (multidimensional arrays of numbers), their outer product is a tensor. The outer product of tensors is also referred to as their tensor product, and can be used to define the tensor algebra.

## Commutative algebra 20 Tensor products review

**24 related questions found**

### What exactly is a tensor?

In simple terms, a tensor is **a dimensional data structure**. Vectors are one-dimensional data structures and matrices are two-dimensional data structures. ... For instance, we can represent second-rank tensors as matrices. This stress on "can be" is important because tensors have properties that not all matrices will have.

### How do you write a tensor product?

If x, y are vectors of length M and N, respectively, their tensor product x⊗y is defined as the M ×N-matrix defined by (x ⊗ y)ij = xiyj. In other words, **x ⊗ y = xyT** . In particular x ⊗ y is a matrix of rank 1, which means that most matrices cannot be written as tensor products.

### What is a pure tensor?

Pure tensor

A pure tensor of V ⊗ W is one that **is of the form v ⊗ w**. It could be written dyadically a^{i}b^{j}, or more accurately a^{i}b^{j} e_{i} ⊗ f_{j}, where the e_{i} are a basis for V and the f_{j} a basis for W. Therefore, unless V and W have the same dimension, the array of components need not be square.

### Does order matter tensor product?

|ϕ(1)⟩⊗|χ(2)⟩ is a cumbersome notation to write ket corresponding to ψ function ϕ(r1)χ(r2), where ri refers to coordinates of the i-th subsystem. That's why **the order of factors in ⊗ product does not matter**; the resulting ket corresponds to the same ψ function and is thus the same ket.

### Is kronecker product commutative?

Kronecker product **is not commutative**, i.e., usually A ⊗ B ≠ B ⊗ A .

### What does tensor product represent?

Tensor Products are used to **describe systems consisting of multiple subsystems**. Each subsystem is described by a vector in a vector space (Hilbert space). For example, let us have two systems I and II with their corresponding Hilbert spaces H_{I} and H_{II}.

### Is kronecker product same as tensor product?

The Kronecker product of matrices corresponds to the **abstract tensor product of linear maps**.

### Is direct product commutative?

The direct product is **commutative and associative up to isomorphism**. That is, G × H ≅ H × G and (G × H) × K ≅ G × (H × K) for any groups G, H, and K. The order of a direct product G × H is the product of the orders of G and H: ... This follows from the formula for the cardinality of the cartesian product of sets.

### Is the tensor product space a vector space?

The tensor product of two vector spaces is a **new vector space** with the property that bilinear maps out of the Cartesian product of the two spaces are equivalently linear maps out of the tensor product.

### Do tensors form a vector space?

The tensors of a given type, with the addition and scalar multiplication inherited from V , form a **vector space on K**. ... sometimes called simple tensors.

### Can you multiply tensors?

Multiplying a Tensor and a Matrix

The **product is calculated by multiplying each mode-n fibre by the U matrix**. Thus, the n-mode product of a tensor with a matrix yields a new tensor. You really do not have to worry about manually calculating n-mode product.

### Which domain in mathematics is called as tensors?

In mathematics, a tensor is **an algebraic object** that describes a multilinear relationship between sets of algebraic objects related to a vector space. ... Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system.

### What are the properties of tensors?

Properties as Tensors: **Physical properties are measured by the interaction of the material with a perturbing driving force**, i.e., a cause. Some physical (thermodynamic) response (effect) can then be measured, and the property defined by the relationship between driving force and response (cause and effect).

### Do tensors commute?

βk is itself a tensor of rank (n+m+j+k) and in no way is a scalar. But their product is **commutative** because the resulting tensor product has the same contravariant and covariant indices.

### What is the difference between tensor and matrix?

In a defined system, a matrix is just a container for entries and it doesn't change if any change occurs in the system, whereas a tensor is an entity in the system that interacts with other entities in a system and **changes its values when other values change**.

### How do you write a tensor product in latex?

The tensor product : **V ⊗ W (Latex: V \otimes W )** . ✒In mathematics, tensor product is itself a vector space.

### What is a tensor in physics?

A tensor is **a concept from mathematical physics that can be thought of as a generalization of a vector**. While tensors can be defined in a purely mathematical sense, they are most useful in connection with vectors in physics. ... In this article, all vector spaces are real and finite-dimensional.

### What is the difference between Cartesian product and tensor product?

In particular, how is it that dimension of Cartesian product is **a sum of dimensions of underlying vector spaces**, while Tensor product, often defined as a quotient of Cartesian product, has dimension which is a product of dimensions of underlying vector spaces.

### Are all vectors tensors?

All vectors are, technically, **tensors**. All tensors are not vectors. This is to say, tensors are a more general object that a vector (strictly speaking though, mathematicians construct tensors through vectors).

### What is tensor example?

A tensor field has a tensor corresponding to each point space. An example is **the stress on a material, such as a construction beam in a bridge**. Other examples of tensors include the strain tensor, the conductivity tensor, and the inertia tensor.