Search for a tool
Kronecker Product

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

Results

Kronecker Product -

Tag(s) : Matrix

Share dCode and you

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!

Please, check our community Discord for help requests!

Thanks to your feedback and relevant comments, dCode has developped the best Kronecker Product tool, so feel free to write! Thank you !

# Kronecker Product

## Kronecker Product

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

### How to multiply 2 matrices with Kronecker?

For $M_1=[a_{ij}]$ a matrix 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 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, $M_1 \otimes M_2 \neq M_1 \dot M_2$

### What are matrix Kronecker multiplication properties?

The Kronecker product suport 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: $\operatorname{det}( A \otimes B ) = \operatorname{det}( A )^{m} \operatorname{det}( B )^{n}$

## Source code

dCode retains ownership of the online 'Kronecker Product' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (PHP, Java, C#, Python, Javascript, Matlab, etc.) no data, script or API access will be for free, same for Kronecker Product download for offline use on PC, tablet, iPhone or Android !

## Need Help ?

Please, check our community Discord for help requests!