克罗内克积是两个任意大小的矩阵间的运算，它是张量积的特殊形式。给定两个矩阵 A ∈ R m × n \boldsymbol{A} \in \mathbb{R}^{m \times n} A∈Rm×n和 B ∈ R p × q \boldsymbol{B} \in \mathbb{R}^{p \times q} B∈Rp×q，则这两个矩阵的克罗内克积是一个在空间 R m p × n q \mathbb{R}^{mp \times nq} Rmp×nq的分块矩阵 A ⊗ B = [ a 11 B ⋯ a 1 n B ⋮ ⋱ ⋮ a m 1 B ⋯ a m n B ] \boldsymbol{A}\otimes\boldsymbol{B}=\left[\begin{array}{ccc}a_{11}\boldsymbol{B}&\cdots & a_{1n}\boldsymbol{B}\\\vdots&\ddots&\vdots\\a_{m1}\boldsymbol{B}&\cdots&a_{mn}\boldsymbol{B}\end{array}\right] A⊗B=⎣⎢⎡​a11​B⋮am1​B​⋯⋱⋯​a1n​B⋮amn​B​⎦⎥⎤​更具体的可以表示为 A ⊗ B = [ a 11 b 11 a 11 b 12 ⋯ a 11 b 1 q ⋯ ⋯ a 1 n b 11 a 1 n b 12 ⋯ a 1 n b 1 q a 11 b 21 a 11 b 22 ⋯ a 11 b 2 q ⋯ ⋯ a 1 n b 21 a 1 n b 22 ⋯ a 1 n b 2 q ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ ⋱ ⋮ a 11 b p 1 a 11 b p 2 ⋯ a 11 b p q ⋯ ⋯ a 1 n b p 1 a 1 n b p 2 ⋯ a 1 n b p q ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ a m 1 b 11 a m 1 b 12 ⋯ a m 1 b 1 q ⋯ ⋯ a m n b 11 a m n b 12 ⋯ a m n b 1 q a m 1 b 21 a m 1 b 22 ⋯ a m 1 b 2 q ⋯ ⋯ a m n b 21 a m n b 22 ⋯ a m n b 2 q ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ ⋱ ⋮ a m 1 b p 1 a m 1 b p 2 ⋯ a m 1 b p q ⋯ ⋯ a m n b p 1 a m n b p 2 ⋯ a m n b p q ] \boldsymbol{A} \otimes \boldsymbol{B}=\left[\begin{array}{ccccccccc} a_{11} b_{11} & a_{11} b_{12} & \cdots & a_{11} b_{1 q} & \cdots & \cdots & a_{1 n} b_{11} & a_{1 n} b_{12} & \cdots & a_{1 n} b_{1 q} \\ a_{11} b_{21} & a_{11} b_{22} & \cdots & a_{11} b_{2 q} & \cdots & \cdots & a_{1 n} b_{21} & a_{1 n} b_{22} & \cdots & a_{1 n} b_{2 q} \\ \vdots & \vdots & \ddots & \vdots & & & \vdots & \vdots & \ddots & \vdots \\ a_{11} b_{p 1} & a_{11} b_{p 2} & \cdots & a_{11} b_{p q} & \cdots & \cdots & a_{1 n} b_{p 1} & a_{1 n} b_{p 2} & \cdots & a_{1 n} b_{p q} \\ \vdots & \vdots & & \vdots & \ddots & & \vdots & \vdots & & \vdots \\ \vdots & \vdots & & \vdots & & \ddots & \vdots & \vdots & & \vdots \\ a_{m 1} b_{11} & a_{m 1} b_{12} & \cdots & a_{m 1} b_{1 q} & \cdots & \cdots & a_{m n} b_{11} & a_{m n} b_{12} & \cdots & a_{m n} b_{1 q} \\ a_{m 1} b_{21} & a_{m 1} b_{22} & \cdots & a_{m 1} b_{2 q} & \cdots & \cdots & a_{m n} b_{21} & a_{m n} b_{22} & \cdots & a_{m n} b_{2 q} \\ \vdots & \vdots & \ddots & \vdots & & & \vdots & \vdots & \ddots & \vdots \\ a_{m 1} b_{p 1} & a_{m 1} b_{p 2} & \cdots & a_{m 1} b_{p q} & \cdots & \cdots & a_{m n} b_{p 1} & a_{m n} b_{p 2} & \cdots & a_{m n} b_{p q} \end{array}\right] A⊗B=⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎡​a11​b11​a11​b21​⋮a11​bp1​⋮⋮am1​b11​am1​b21​⋮am1​bp1​​a11​b12​a11​b22​⋮a11​bp2​⋮⋮am1​b12​am1​b22​⋮am1​bp2​​⋯⋯⋱⋯⋯⋯⋱⋯​a11​b1q​a11​b2q​⋮a11​bpq​⋮⋮am1​b1q​am1​b2q​⋮am1​bpq​​⋯⋯⋯⋱⋯⋯⋯​⋯⋯⋯⋱⋯⋯⋯​a1n​b11​a1n​b21​⋮a1n​bp1​⋮⋮amn​b11​amn​b21​⋮amn​bp1​​a1n​b12​a1n​b22​⋮a1n​bp2​⋮⋮amn​b12​amn​b22​⋮amn​bp2​​⋯⋯⋱⋯⋯⋯⋱⋯​a1n​b1q​a1n​b2q​⋮a1n​bpq​⋮⋮amn​b1q​amn​b2q​⋮amn​bpq​​⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎤​

 克罗内克积满足如下性质：

[ 1 2 3 1 ] ⊗ [ 0 3 2 1 ] = [ 1 ⋅ 0 1 ⋅ 3 2 ⋅ 0 2 ⋅ 3 1 ⋅ 2 1 ⋅ 1 2 ⋅ 2 2 ⋅ 1 3 ⋅ 0 3 ⋅ 3 1 ⋅ 0 1 ⋅ 3 3 ⋅ 2 3 ⋅ 1 1 ⋅ 2 1 ⋅ 1 ] = [ 0 3 0 6 2 1 4 2 0 9 0 3 6 3 2 1 ] \left[\begin{array}{ll} 1 & 2 \\ 3 & 1 \end{array}\right] \otimes\left[\begin{array}{ll} 0 & 3 \\ 2 & 1 \end{array}\right]=\left[\begin{array}{llll} 1 \cdot 0 & 1 \cdot 3 & 2 \cdot 0 & 2 \cdot 3 \\ 1 \cdot 2 & 1 \cdot 1 & 2 \cdot 2 & 2 \cdot 1 \\ 3 \cdot 0 & 3 \cdot 3 & 1 \cdot 0 & 1 \cdot 3 \\ 3 \cdot 2 & 3 \cdot 1 & 1 \cdot 2 & 1 \cdot 1 \end{array}\right]=\left[\begin{array}{llll} 0 & 3 & 0 & 6 \\ 2 & 1 & 4 & 2 \\ 0 & 9 & 0 & 3 \\ 6 & 3 & 2 & 1 \end{array}\right] [13​21​]⊗[02​31​]=⎣⎢⎢⎡​1⋅01⋅23⋅03⋅2​1⋅31⋅13⋅33⋅1​2⋅02⋅21⋅01⋅2​2⋅32⋅11⋅31⋅1​⎦⎥⎥⎤​=⎣⎢⎢⎡​0206​3193​0402​6231​⎦⎥⎥⎤​

[ a 11 a 12 a 21 a 22 a 31 a 32 ] ⊗ [ b 11 b 12 b 13 b 21 b 22 b 23 ] = [ a 11 b 11 a 11 b 12 a 11 b 13 a 12 b 11 a 12 b 12 a 12 b 13 a 11 b 21 a 11 b 22 a 11 b 23 a 12 b 21 a 12 b 22 a 12 b 23 a 21 b 11 a 21 b 12 a 21 b 13 a 22 b 11 a 22 b 12 a 22 b 13 a 21 b 21 a 21 b 22 a 21 b 23 a 22 b 21 a 22 b 22 a 22 b 23 a 31 b 11 a 31 b 12 a 31 b 13 a 32 b 11 a 32 b 12 a 32 b 13 a 31 b 21 a 31 b 22 a 31 b 23 a 32 b 21 a 32 b 22 a 32 b 23 ] \left[\begin{array}{l} a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \end{array}\right] \otimes\left[\begin{array}{lll} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{array}\right]=\left[\begin{array}{llllll} a_{11} b_{11} & a_{11} b_{12} & a_{11} b_{13} & a_{12} b_{11} & a_{12} b_{12} & a_{12} b_{13} \\ a_{11} b_{21} & a_{11} b_{22} & a_{11} b_{23} & a_{12} b_{21} & a_{12} b_{22} & a_{12} b_{23} \\ a_{21} b_{11} & a_{21} b_{12} & a_{21} b_{13} & a_{22} b_{11} & a_{22} b_{12} & a_{22} b_{13} \\ a_{21} b_{21} & a_{21} b_{22} & a_{21} b_{23} & a_{22} b_{21} & a_{22} b_{22} & a_{22} b_{23} \\ a_{31} b_{11} & a_{31} b_{12} & a_{31} b_{13} & a_{32} b_{11} & a_{32} b_{12} & a_{32} b_{13} \\ a_{31} b_{21} & a_{31} b_{22} & a_{31} b_{23} & a_{32} b_{21} & a_{32} b_{22} & a_{32} b_{23} \end{array}\right] ⎣⎡​a11​a21​a31​​a12​a22​a32​​⎦⎤​⊗[b11​b21​​b12​b22​​b13​b23​​]=⎣⎢⎢⎢⎢⎢⎢⎡​a11​b11​a11​b21​a21​b11​a21​b21​a31​b11​a31​b21​​a11​b12​a11​b22​a21​b12​a21​b22​a31​b12​a31​b22​​a11​b13​a11​b23​a21​b13​a21​b23​a31​b13​a31​b23​​a12​b11​a12​b21​a22​b11​a22​b21​a32​b11​a32​b21​​a12​b12​a12​b22​a22​b12​a22​b22​a32​b12​a32​b22​​a12​b13​a12​b23​a22​b13​a22​b23​a32​b13​a32​b23​​⎦⎥⎥⎥⎥⎥⎥⎤​