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 specialization of the tensor 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. The misattribution to Kronecker rather than Zehfuss was due to Kurt Hensel. If A is an m × n matrix and B is a p × q matrix, then the Kronecker product A ⊗ B is the pm × qn block matrix: more explicitly: Using and to denote truncating integer division and remainder, respectively, and numbering the matrix elements starting from 0, one obtains and For the usual numbering starting from 1, one obtains and If A and B represent linear transformations V1 → W1 and V2 → W2, respectively, then the tensor product of the two maps is represented by A ⊗ B, which is the same as V1 ⊗ V2 → W1 ⊗ W2. Similarly: The Kronecker product can be used to get a convenient representation for some matrix equations. Consider for instance the equation AXB = C, where A, B and C are given matrices and the matrix X is the unknown. We can use the "vec trick" to rewrite this equation as Here, vec(X) denotes the vectorization of the matrix X, formed by stacking the columns of X into a single column vector. It now follows from the properties of the Kronecker product that the equation AXB = C has a unique solution, if and only if A and B are invertible .

About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Ontological neighbourhood
Related courses (25)
PHYS-432: Quantum field theory II
The goal of the course is to introduce relativistic quantum field theory as the conceptual and mathematical framework describing fundamental interactions such as Quantum Electrodynamics.
MATH-111(e): Linear Algebra
L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications.
MATH-646: Reading group in quantum computing
Quantum computing has received wide-spread attention lately due the possibility of a near-term breakthrough of quantum supremacy. This course acts as an introduction to the area of quantum computing.
Show more
Related publications (33)
Related MOOCs (9)
Algebra (part 1)
Un MOOC francophone d'algèbre linéaire accessible à tous, enseigné de manière rigoureuse et ne nécessitant aucun prérequis.
Algebra (part 1)
Un MOOC francophone d'algèbre linéaire accessible à tous, enseigné de manière rigoureuse et ne nécessitant aucun prérequis.
Algebra (part 2)
Un MOOC francophone d'algèbre linéaire accessible à tous, enseigné de manière rigoureuse et ne nécessitant aucun prérequis.
Show more

Graph Chatbot

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.