Kronecker product

The Kronecker product, denoted by \otimes, is a multiplication method for generating a new vector space from existing vector spaces, and therefore, new vectors from existing vectors.

Consider 2 vectors spaces, e.g. V=\mathbb{R}^{2} and W=\mathbb{R}^{3}. For \boldsymbol{\mathit{v}}=\begin{pmatrix} a_1\\a_2 \end{pmatrix} in V and \boldsymbol{\mathit{w}}=\begin{pmatrix} b_1\\b_2\\b_3 \end{pmatrix} in W, we can define a new vector space, V\otimes W, which consists of the vector \boldsymbol{\mathit{v}}\otimes\boldsymbol{\mathit{w}}, where:

\boldsymbol{\mathit{v}}\otimes\boldsymbol{\mathit{w}}=\begin{pmatrix} a_1\\a_2 \end{pmatrix}\otimes \begin{pmatrix} b_!\\b_2 \\ b_3 \end{pmatrix}=\begin{pmatrix} a_1b_1\\a_1b_2 \\a_1b_3 \\ a_2b_1 \\ a_2b_2 \\ a_2b_3 \end{pmatrix}

If the basis vectors for V and W are V=\left \{\boldsymbol{\mathit{e_1}},\boldsymbol{\mathit{e_2}}\right \} and W=\left \{\boldsymbol{\mathit{f_1}},\boldsymbol{\mathit{f_2}},\boldsymbol{\mathit{f_3}}\right \} respectively, the basis for V\otimes W is:



Why is a new vector space?


An -dimensional vector space is spanned by  linearly independent basis vectors. The basis vectors for V=\left \{\boldsymbol{\mathit{e_1}},\boldsymbol{\mathit{e_2}}\right \} and W=\left \{\boldsymbol{\mathit{f_1}},\boldsymbol{\mathit{f_2}},\boldsymbol{\mathit{f_3}}\right \} are

and consequently, the basis vectors for  are

These 6 linearly independent basis vectors therefore span a 6-dimensional space.


This implies that V\otimes W is nm dimensional if V is n-dimensional and W is m-dimensional. Since V\otimes W is a vector space, the vectors \boldsymbol{\mathit{v}}\otimes\boldsymbol{\mathit{w}} must follow the rules of addition and multiplication of a vector space. Each vector \boldsymbol{\mathit{v}}\otimes\boldsymbol{\mathit{w}} in the new vector space can then be written as a linear combination of the basis vectors \boldsymbol{\mathit{e_i}}\otimes\boldsymbol{\mathit{f_j}}, i.e. \sum c_{i,j}\boldsymbol{\mathit{e_i}}\otimes\boldsymbol{\mathit{f_j}}.

In general, if


Since the pair  in  is distinct for each \boldsymbol{\mathit{e_i}}\otimes\boldsymbol{\mathit{f_j}} vector, the Kronecker product \boldsymbol{\mathit{e_i}}\otimes\boldsymbol{\mathit{f_j}} results in  basis vectors, which span an  vector space.

As mentioned in an earlier article, a vector space is a set of objects that follows certain rules of addition and multiplication. If the objects are matrices, we have a vector space of matrices. For example, the vector spaces of matrices and generates a new vector space of matrices , where

Similarly, if the objects are functions, we have a vector space of functions and the Kronecker product of two vector spaces of functions  and generates a new vector space of functions . If and are spanned by basis functions and basis functions respectively, is spanned by basis functions.

A vector space that is generated from two separate vector spaces has applications in quantum composite systems and in group theory.


What is the relation between the matrix entries of A, B and C in ?


Let the matrix entries of A, B and C be , and respectively, where

Using the ordering convention called dictionary order, where  is determined by  and , and is determined by and , such that  and are given by

For example, if and ,

We can then express the matrix entries of as .



Next article: operators
Previous article: hilbert space
Content page of quantum mechanics
Content page of advanced chemistry
Main content page

Leave a Reply

Your email address will not be published. Required fields are marked *