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  "body": "\n## Definition\n\nLet $F$ be a [[field|04C5ZBG1WT]] and let $A$ be an $m \\times n$ [[matrix|0VD9ZSWWJG]] over $F$.\nA *generalized inverse* of $A$ is an $n \\times m$ matrix $G$ over $F$ such that\n$$A G A = A.$$\n\n## Remarks\n\nExample: for $A = \\begin{pmatrix} 1 \u0026 1 \\end{pmatrix}$, the column $G = \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix}$ is a generalized inverse, because $AG = (1)$ and so $AGA = A$.\nNon-example: for $A = \\begin{pmatrix} 1 \u0026 0 \\\\ 0 \u0026 0 \\end{pmatrix}$, the matrix $G = \\begin{pmatrix} 2 \u0026 0 \\\\ 0 \u0026 0 \\end{pmatrix}$ is not one, because $AGA = \\begin{pmatrix} 2 \u0026 0 \\\\ 0 \u0026 0 \\end{pmatrix} \\ne A$.\nIf $A$ is square and [[invertible|0EB9N0S33C]], then $A^{-1}$ is its only generalized inverse.\nIf $Ax = b$ has a solution and $G$ is a generalized inverse, then $x = Gb$ is a solution, since $AGb = AGAx = Ax = b$.\nA generalized inverse is often written $A^-$ or $A^{(1)}$; it is also called a $\\{1\\}$-inverse.\nA matrix $G$ that satisfies also $GAG = G$ is a reflexive generalized inverse.\n",
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