What is Kronecker product: Definition and 12 Discussions

In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to matrices, and gives the matrix of the tensor product linear map with respect to a standard choice of basis. The Kronecker product is to be distinguished from the usual matrix multiplication, which is an entirely different operation. The Kronecker product is also sometimes called matrix direct product.The Kronecker product is named after the German mathematician Leopold Kronecker (1823–1891), even though there is little evidence that he was the first to define and use it. The Kronecker product has also been called the Zehfuss matrix, and the Zehfuss product, after Johann Georg Zehfuss, who in 1858 described this matrix operation, but Kronecker product is currently the most widely used.

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  1. Dr_Nate

    I Composite quantum systems: Kronecker and Hadamard/Schur products

    In QM textbooks, authors will often jam two kets next to each other and say nothing about the binary operation between them. Other times, it may be called a tensor product, Kronecker product, direct product, or, in Griffith's case, a simple product. I ask the following question in this forum...
  2. S

    I What is the proper matrix product?

    It says in any textbook (for example, in classical text «Theory of matrices» by P. Lankaster) on matrix theory that matrices form an algebra with the following obvious operations: 1) matrix addition; 2) multiplication by the undelying field elements; 3) matrix multiplication. Is the last one...
  3. Theta_84

    Can Einstein Index Notation Help Me Solve Equations in Continuum Mechanics?

    Hello I am doing some exercises in continuum mechanics and it is a little bit confusing. I am given the following equations ## A_{ij}= \delta_{ij} +au_{i}v_{j} ## and ## (A_{ij})^{-1} = \delta_{ij} - \frac{au_{i}v_{j}}{1-au_{k}v_{k}}##. If I want to take the product to verify that they give...
  4. arivero

    I Kronecker product recovering the initial representation?

    In E6, the product 27 x 27 contains the (conjugate) 27. In SU(3), something similar happens with 3 x 3, which decomposes as 3 + 6. I was wondering, how usual is this? Do we have some lemmas telling when a product N x N is going to "recover" the original N, or its conjugate, inside the sum?
  5. E

    Kronecker sum of more than two matrices?

    Homework Statement The question arises from this quote from wikipedia's article about kronecker product: Kronecker sums appear naturally in physics when considering ensembles of non-interacting systems. Let Hi be the Hamiltonian of the i-th such system. Then the total Hamiltonian of the...
  6. gfd43tg

    Compute Kronecker Product of Two Arrays in MATLAB - Step-by-Step Guide

    Homework Statement In this problem, you will write code that computes the Kronecker product of two arrays. Suppose A is a numeric array of size r-by-c and B is a numeric array of size n-by-m. Then the Kronecker product of A with B is a numeric array, of dimension rn-by-cm, defined as:Homework...
  7. U

    Can B Be Determined from Given A and C in Kronecker Product?

    hi everybody Today I have a question about Kronecker products, If you have a direct answer it is perfect but if not, any kind of paper reference might work as well. now say I have to matrices A and B in general there is nothing special about them. They are not hermitian or triangular or what...
  8. Y

    Kronecker product of infinite dimensional matrices

    Hi there, I was recently working with Kronecker product of matrices, and a question came up that I'm not sure how to answer. Is the matrix that represents a Kronecker product of two infinite dimensional matrices well defined? If yes, are some of the properties of the Kronecker product listed in...
  9. A

    Computing a kronecker product of two matrices

    Does anyone know an algorithm for computing kronecker products of two matrices? It's probably not that hard, but I feel like my head is about to explode ATM, so if you can help me out that'd be cool. I want to implement this in fortran... I'll give you an example; Say I want compute the...
  10. S

    Kronecker Product: Solving AA` with w & D

    Hi everyone, Please help me with this problem. Suppose w be a n x n symmetric matrix and D be n x m matrix. Let A=wDD`w. Is it possible to write the matrix, AA`= (wDD`w)(wDD`w) as the kronecker product of any two matrices? Thanks in advance.
  11. P

    How to Align Kronecker Product Results with Scalar Elements in a Matrix?

    The Kronecker product of an argument X and a 2x2 matrix, increases the dimensions of each argument X individually. If each argument X is a scalar value, it now becomes a 2x2 matrix. How are these arguments now aligned with each other and the other elements in the resultant matrix? For...
  12. C

    The (i,j) component of a Kronecker product?

    By the Kronecker product I mean the ordinary tensor product of matrices. In my case I am only interested in square matrices, in fact I want to compute the nonzero elements of products like XXZZXXZZ where X and Z are 2x2 matrices (in fact they are the pauli matrices e.g. the standard...