Kronecker Product#
Definition: Kronecker Product
Let \(A = (a_{ij}) \in F^{m \times n}\) be an \(m \times n\)-matrix and let \(B = (b_{ij}) \in F^{p \times q}\) be a \(p \times q\)-matrix.
The Kronecker product of \(A\) with \(B\) is a \(pm \times nq\)-matrix defined as
\[ \begin{bmatrix} a_{11} B & \cdots & a_{1n} B \\ \vdots & \ddots & \vdots \\ a_{m1} B & \cdots & a_{mn} B \end{bmatrix} \]
Notation
The Kronecker product is written in one of the following ways:
\[A \otimes B \qquad A \otimes_{\text{Kron}} B\]
Example
\[\begin{aligned}\begin{bmatrix}1 & 2 & 3 \\ 4 & 5 & 6\end{bmatrix} \otimes \begin{bmatrix} -1 & -2 & -3 \\ -4 & -5 & -6 \\ -7 & -8 & -9 \end{bmatrix} &= \begin{bmatrix} 1 \cdot \begin{bmatrix} -1 & -2 & -3 \\ -4 & -5 & -6 \\ -7 & -8 & -9 \end{bmatrix} & 2 \cdot \begin{bmatrix} -1 & -2 & -3 \\ -4 & -5 & -6 \\ -7 & -8 & -9 \end{bmatrix} & 3 \cdot \begin{bmatrix} -1 & -2 & -3 \\ -4 & -5 & -6 \\ -7 & -8 & -9 \end{bmatrix} \\ 4 \cdot \begin{bmatrix} -1 & -2 & -3 \\ -4 & -5 & -6 \\ -7 & -8 & -9 \end{bmatrix} & 5 \cdot \begin{bmatrix} -1 & -2 & -3 \\ -4 & -5 & -6 \\ -7 & -8 & -9 \end{bmatrix} & 6 \cdot \begin{bmatrix} -1 & -2 & -3 \\ -4 & -5 & -6 \\ -7 & -8 & -9 \end{bmatrix} \end{bmatrix} \\ &= \begin{bmatrix} 1 \cdot (-1) & 1 \cdot (-2) & 1 \cdot (-3) & 2 \cdot (-1) & 2 \cdot (-2) & 2 \cdot (-3) & 3 \cdot (-1) & 3 \cdot (-2) & 3 \cdot (-3) \\ 1 \cdot (-4) & 1 \cdot (-5) & 1 \cdot (-6) & 2 \cdot (-4) & 2 \cdot (-4) & 2 \cdot (-6) & 3 \cdot (-4) & 3 \cdot (-5) & 3 \cdot (-6) \\ 1 \cdot (-7) & 1 \cdot (-8) & 1 \cdot (-9) & 2 \cdot (-7) & 2 \cdot (-8) & 2 \cdot (-9) & 3 \cdot (-7) & 3 \cdot (-8) & 3 \cdot (-9) \\ 4 \cdot (-1) & 4 \cdot (-2) & 4 \cdot (-3) & 5 \cdot (-1) & 5 \cdot (-2) & 5 \cdot (-3) & 6 \cdot (-1) & 6 \cdot (-2) & 6 \cdot (-3) \\ 4 \cdot (-4) & 4 \cdot (-5) & 4 \cdot (-6) & 5 \cdot (-4) & 5 \cdot (-4) & 5 \cdot (-6) & 6 \cdot (-4) & 6 \cdot (-5) & 6 \cdot (-6) \\ 4 \cdot (-7) & 4 \cdot (-8) & 4 \cdot (-9) & 5 \cdot (-7) & 5 \cdot (-8) & 5 \cdot (-9) & 6 \cdot (-7) & 6 \cdot (-8) & 6 \cdot (-9) \end{bmatrix} \\ &= \begin{bmatrix} -1 & -2 & -3 & -2 & -4 & -6 & -3 & -6 & -9 \\ -4 & -5 & -6 & -8 & -10 & -12 & -12 & -15 & -18 \\ -7 & -8 & -9 & -14 & -16 & -18 & -21 & -24 & -27 \\ -4 & -8 & -12 & -5 & -10 & -15 & -6 & -12 & -18 \\ -16 & -20 & -24 & -20 & -25 & -30 & -24 & -30 & -36 \\ -28 & -32 & -36 & -35 & -40 & -45 & -42 & -48 & -54 \end{bmatrix} \end{aligned} \]
Theorem: Associativity of the Kronecker Product
The Kronecker product is associative:
\[(A \otimes B) \otimes C = A \otimes (B \otimes C)\]
Proof
TODO
Theorem: Bilinearity of the Kronecker Product
The Kronecker product is bilinear:
\[\begin{aligned}A \otimes (\lambda B + \mu C) &= \lambda \cdot (A \otimes B) + \mu \cdot (A \otimes C) \\ (\lambda A + \mu B) \otimes C &= \lambda \cdot (A \otimes C) + \mu \cdot (B \otimes C) \end{aligned}\]
Proof
TODO
Theorem: Mixed-Product Property
The Kronecker product has the following property whenever the matrix products \(AC\) and \(BD\) are possible:
\[(A \otimes B) (C \otimes D) = (AC) \otimes (BD)\]
Proof
TODO