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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be a [[field|04C5ZBG1WT]] and let $m, n \\ge 1$.\nLet $A$ be an $m \\times m$ [[matrix|0VD9ZSWWJG]], $B$ an $n \\times n$ matrix and $C$ an $m \\times n$ matrix over $\\mathbb{K}$.\nThe *Sylvester equation* for $A$, $B$, $C$ is the equation\n$$AX + XB = C$$\nfor an unknown $m \\times n$ matrix $X$ over $\\mathbb{K}$.\n\n## Remarks\n\nThe map $X \\mapsto AX + XB$ is linear, and with the [[Kronecker product|0N7ENQRCXN]] the equation reads $(I_n \\otimes A + B^T \\otimes I_m) \\operatorname{vec}(X) = \\operatorname{vec}(C)$.\nIt has a unique solution for every $C$ if and only if $A$ and $-B$ have no common [[eigenvalue|0BQAA2BK3Q]] in an algebraic closure of $\\mathbb{K}$.\nExample: $A = \\operatorname{diag}(1, 2)$, $B = 1$ and $C = \\begin{pmatrix} 2 \\\\ 3 \\end{pmatrix}$ give $x_1 + x_1 = 2$ and $2x_2 + x_2 = 3$, so $X = \\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix}$.\nNon-example: the equation $AXB = C$ is not a Sylvester equation, since $X$ is multiplied on both sides in one term.\nFor $B = -A$, the equation $AX - XA = C$ never has a unique solution, because $A$ and $A$ share their eigenvalues.\n",
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