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

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

For \( M_1=[a_{ij}] \) 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 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** 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: \( \operatorname{det}( A \otimes B ) = \operatorname{det}( A )^{m} \operatorname{det}( B )^{n} \)

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