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  "body": "\n## Definition\n\nLet $F$ be a [[field|04C5ZBG1WT]] and let $A$ be an $n \\times n$ [[square matrix|0JXZKX526R]] over $F$.\nThe *index* of $A$ is the least integer $k \\ge 0$ such that $\\rank(A^{k+1}) = \\rank(A^k)$, where $\\rank$ is the [[rank|0ZC9YTXZRY]] and $A^0 = I$.\nThe *Drazin inverse* of $A$ is the $n \\times n$ matrix $A^D$ over $F$ such that, with $k$ the index of $A$:\n\n1. $A^{k+1} A^D = A^k$.\n2. $A^D A A^D = A^D$.\n3. $A A^D = A^D A$.\n\nThere is exactly one matrix with these three properties.\n\n## Remarks\n\nExample: if $A$ is invertible, its index is $0$, and $A^D = A^{-1}$; indeed $A \\cdot A^{-1} = I$, $A^{-1} A A^{-1} = A^{-1}$, and $A^{-1}$ commutes with $A$.\nExample: if $A$ is [[nilpotent|07BSPZPSMF]] with $A^k = 0$ for its index $k$, then $A^D = 0$; each side of each condition is then $0$, because $A^k = 0$.\nNon-example: for $A = \\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$, of index $2$, the matrix $X = \\begin{pmatrix} 0 \u0026 0 \\\\ 1 \u0026 0 \\end{pmatrix}$ is not $A^D$, because $AX = \\begin{pmatrix} 1 \u0026 0 \\\\ 0 \u0026 0 \\end{pmatrix}$ and $XA = \\begin{pmatrix} 0 \u0026 0 \\\\ 0 \u0026 1 \\end{pmatrix}$ differ, so clause 3 fails.\nIf $A = P \\begin{pmatrix} C \u0026 0 \\\\ 0 \u0026 N \\end{pmatrix} P^{-1}$ with $C$ invertible and $N$ nilpotent, then $A^D = P \\begin{pmatrix} C^{-1} \u0026 0 \\\\ 0 \u0026 0 \\end{pmatrix} P^{-1}$.\nWhen the index is at most $1$, the Drazin inverse is called the group inverse and written $A^{\\#}$.\n",
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