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  "body": "\n## Definition\n\nLet $A$ be an $m \\times n$ complex matrix.\nThere is exactly one $n \\times m$ complex matrix $A^+$ with these four properties:\n\n1. $A A^+ A = A$.\n2. $A^+ A A^+ = A^+$.\n3. $(A A^+)^* = A A^+$.\n4. $(A^+ A)^* = A^+ A$.\n\nThis matrix is the *Moore–Penrose inverse*, or *pseudoinverse*, of $A$, where $^*$ is the [[conjugate transpose|0DXKE10PHF]].\n\n## Remarks\n\nIf $A$ is invertible, then $A^+ = A^{-1}$.\nIf the columns of $A$ are linearly independent, then $A^+ = (A^* A)^{-1} A^*$.\nFor $b \\in \\C^m$, the vector $A^+ b$ is the least squares solution of $Ax = b$ with the least norm.\nIf $A = U \\Sigma V^*$ is a singular value decomposition, then $A^+ = V \\Sigma^+ U^*$, where $\\Sigma^+$ inverts the nonzero singular values and transposes.\n\n## Explanations\n\n### L1 human\n\nThe pseudoinverse extends the inverse to all matrices, also singular and rectangular ones.\nIt gives the best possible solution of $Ax = b$ when no exact solution exists or when there are many.\nFor the zero matrix, the pseudoinverse is the zero matrix of the transposed size.\n",
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