What is the tensor product of two matrices?

Tensor product of two matrices (by D.A. Suprunenko) Here, A is an (m×n)-matrix, B is a (p×q)-matrix and A⊗B is an (mp×nq)-matrix over an associative commutative ring k with a unit. where α∈k, (A⊗B)(C⊗D)=AC⊗BD.

How do you find the product of a tensor matrix?

We start by defining the tensor product of two vectors. Definition 7.1 (Tensor product of vectors). 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 .

Is kronecker product a tensor product?

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

What does tensor product represent?

In mathematics, the tensor product of representations is a tensor product of vector spaces underlying representations together with the factor-wise group action on the product.

How do you write a tensor product?

The first is a vector (v,w) in the direct sum V⊕W V ⊕ W (this is the same as their direct product V×W V × W ); the second is a vector v⊗w v ⊗ w in the tensor product V⊗W V ⊗ W . And that’s it! 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 .

Are tensor product and outer product the same?

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.

Is tensor product a tensor?

Something that behaves the right way under certain changes of variables is a tensor. And then there’s things that aren’t called tensors, but they have tensor products. These seem simple enough in some cases—you think “I didn’t realize that has a name. So it’s called a tensor product.

Is the tensor product distributive?

Also, the tensor product obeys a distributive law with the direct sum operation: . The definition is the same no matter which scalar field is used. Using tensor products, one can define symmetric tensors, antisymmetric tensors, as well as the exterior algebra.

What is a tensor product in quantum mechanics?

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 HI and HII.

Is a tensor A matrix?

A tensor is a container which can house data in N dimensions. Often and erroneously used interchangeably with the matrix (which is specifically a 2-dimensional tensor), tensors are generalizations of matrices to N-dimensional space. Mathematically speaking, tensors are more than simply a data container, however.

What is tensor product surface?

Tensor product surfaces. • Natural way to think of a surface: curve is swept, and (possibly) deformed. Examples: ruled surface (line is swept), surface of revolution (circle is swept along line, grows and shrinks).