Linear and multilinear algebra; matrix theory Definition

Kronecker product

Definition

Let $F$ be a field, let $A = (a_{ij})$ be an $m \times n$ matrix over $F$ , and let $B$ be a $p \times q$ matrix over $F$ . The Kronecker product $A \otimes B$ is the $mp \times nq$ block matrix $A \otimes B = \begin{pmatrix} a_{11} B & \cdots & a_{1n} B \\ \vdots & \ddots & \vdots \\ a_{m1} B & \cdots & a_{mn} B \end{pmatrix}.$

Remarks

It satisfies $(A \otimes B)(C \otimes D) = (AC) \otimes (BD)$ when the products $AC$ and $BD$ are defined. If $A$ and $B$ are invertible, then $(A \otimes B)^{-1} = A^{-1} \otimes B^{-1}$ . It is not commutative in general, but $A \otimes B$ and $B \otimes A$ are permutation equivalent. It is the matrix of the tensor product of two linear maps, in the tensor product bases ordered suitably.

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Matrix definition
Block matrix definition
Kronecker product definition

Tag	`0JK6M18QZH`
URI	`m2://0JK6M18QZH`
Kind	definition
Writing	W1 stated
MSC	15A69
Defines	Kronecker product
Uses	Matrix, Block matrix

Kronecker product

Definition

Remarks

Reading path

Further reading

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