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  "body": "\n## Summary\n\nThe Drazin inverse of a square matrix $A$ is a matrix $X$ that commutes with $A$ and satisfies $XAX = X$ and $A^{k+1}X = A^k$ for some $k$.\nIt exists and is unique for every square matrix over a field.\nIt inverts $A$ on the part of the space where $A$ acts invertibly, and it is $0$ on the rest.\n\n## Explanations\n\n### L1 human\n\nA square matrix splits the space into two pieces that it keeps: one where it acts invertibly and one where some power of it is zero.\nThe Drazin inverse inverts the matrix on the first piece and is zero on the second.\nFor example, for $A = \\operatorname{diag}(2, 0)$, the Drazin inverse is $\\operatorname{diag}(1/2, 0)$.\nUnlike the [[generalized inverse|FPHNJ5AAYF]], it is defined through powers of $A$, so it fits well with polynomials in $A$.\n",
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