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) : Mathematics,Algebra,Symbolic Computation

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

You have a problem, an idea for a project, a specific need and dCode can not (yet) help you? You need custom development? *Contact-me*!

This page is using the new English version of dCode, *please make comments* !

Sponsored ads

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

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

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

This product is not equivalent to the classical multiplication" target="_blank">matrix product, \( M_1 \otimes M_2 \neq M_1 \dot M_2 \)

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 is is non-commutative

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

It has also some 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: \( \operatorname{det}( A \otimes B ) = \operatorname{det}( A )^{m} \operatorname{det}( B )^{n} \)

dCode retains ownership of the source code of the script Kronecker Product. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, etc.) which dCode owns rights can be transferred after sales quote. So if you need to download the Kronecker Product script for offline use, for you, your company or association, see you on contact page !

kronecker,product,multiplication,matrix,tensor

Source : http://www.dcode.fr/kronecker-product

© 2017 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaches. dCode