Tool to calculate a Kronecker matrix product in computer algebra. The Kronecker product is a special case of tensor multiplication on matrices.

Kronecker Product - dCode

Tag(s) : Matrix

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!

A suggestion ? a feedback ? a bug ? an idea ? *Write to dCode*!

For $ M_1=[a_{ij}] $ a matrix/tensor with $ m $ lines and $ n $ columns and $ M_2=[b_{ij}] $ a matrix with $ p $ lines and $ q $ columns. The Kronecker product is noted with a symbol: a circled cross `⊗`. $ M_1 \otimes M_2 = [c_{ij}] $ is a larger matrix of $ m \times p $ lines and $ n \times q $ columns, with: $$ \forall i, j: c_{ij} = a_{ij}.B $$

__Example:__ $$ M=\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \otimes \begin{bmatrix} 7 & 8 \\ 9 & 10 \end{bmatrix} = \begin{bmatrix} 7 & 8 & 14 & 16 & 21 & 24 \\ 9 & 10 & 18 & 20 & 27 & 30 \\ 28 & 32 & 35 & 40 & 42 & 48 \\ 36 & 40 & 45 & 50 & 54 & 60 \end{bmatrix} $$

This product is not equivalent to the classical multiplication">matrix product, $ M_1 \otimes M_2 \neq M_1 \dot M_2 $

The Kronecker product supports associativity:

$$ A \otimes (B+ \lambda\ \cdot C) = (A \otimes B) + \lambda (A \otimes C) \\ (A + \lambda\ \cdot B) \otimes C = (A \otimes C) + \lambda (B \otimes C) \\ A \otimes ( B \otimes C) = (A \otimes B) \otimes C \\ (A \otimes B) (C \otimes D) = (A C) \otimes (B D) $$

But Kronecker product is non-commutative

$$ A \otimes B \neq B \otimes A $$

Kronecker product has also some distributivity properties:

— Distributivity over matrix transpose: $ ( A \otimes B )^T = A^T \otimes B^T $

— Distributivity over matrix traces: $ \operatorname{Tr}( A \otimes B ) = \operatorname{Tr}( A ) \operatorname{Tr}( B ) $

— Distributivity over matrix determinants: $ \operatorname{det}( A \otimes B ) = \operatorname{det}( A )^{m} \operatorname{det}( B )^{n} $

The name is a tribute to the German mathematician Leopold Kronecker.

dCode retains ownership of the "Kronecker Product" source code. Except explicit open source licence (indicated Creative Commons / free), the "Kronecker Product" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Kronecker Product" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Kronecker Product" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!

Reminder : dCode is free to use.

The copy-paste of the page "Kronecker Product" or any of its results, is allowed (even for commercial purposes) as long as you cite dCode!

Exporting results as a .csv or .txt file is free by clicking on the *export* icon

Cite as source (bibliography):

*Kronecker Product* on dCode.fr [online website], retrieved on 2023-09-21,

kronecker,product,multiplication,matrix,tensor

https://www.dcode.fr/kronecker-product

© 2023 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaching / CTF.

Feedback