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  "body": "\n## Definition\n\nLet $F$ be a field, let $A = (a_{ij})$ be an $m \\times n$ [[matrix|0VD9ZSWWJG]] over $F$, and let $B$ be a $p \\times q$ matrix over $F$.\nThe *Kronecker product* $A \\otimes B$ is the $mp \\times nq$ [[block matrix|0ASDFEQS6P]]\n$$A \\otimes B = \\begin{pmatrix} a_{11} B \u0026 \\cdots \u0026 a_{1n} B \\\\ \\vdots \u0026 \\ddots \u0026 \\vdots \\\\ a_{m1} B \u0026 \\cdots \u0026 a_{mn} B \\end{pmatrix}.$$\n\n## Remarks\n\nIt satisfies $(A \\otimes B)(C \\otimes D) = (AC) \\otimes (BD)$ when the products $AC$ and $BD$ are defined.\nIf $A$ and $B$ are invertible, then $(A \\otimes B)^{-1} = A^{-1} \\otimes B^{-1}$.\nIt is not commutative in general, but $A \\otimes B$ and $B \\otimes A$ are permutation equivalent.\nIt is the matrix of the tensor product of two linear maps, in the tensor product bases ordered suitably.\n\n## Explanations\n\n### L1 human\n\nThe Kronecker product replaces each entry of $A$ with that entry times the whole matrix $B$.\nFor example, $I_2 \\otimes B$ is the block diagonal matrix with two copies of $B$.\nIt is the matrix form of the tensor product.\n",
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