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  "body": "\n## Summary\n\nA Householder transformation is the [[reflection|06BRDF3F0E]] of $\\R^n$ or $\\C^n$ in a hyperplane through the origin.\nIts matrix is $I - 2 v v^*$ for a unit vector $v$ normal to the hyperplane.\nIt is an [[orthogonal|0W02QESPJQ]] or unitary matrix, and in $\\R^n$ it can send any given vector to any other vector of the same length.\nThis makes it the basic tool for creating zeros in a matrix in a stable way.\nIt is used in the [[QR decomposition|0ERWG4NXQ9]] and in the reduction to Hessenberg or tridiagonal form.\n\n## Explanations\n\n### L1 human\n\nThink of a mirror through the origin.\nThe Householder transformation reflects every vector in this mirror: the part along the mirror stays, and the part perpendicular to it flips sign.\nBy placing the mirror halfway between two vectors of the same length, one can swap them.\nFor example, the mirror between $(3, 4)$ and $(5, 0)$ in the plane sends $(3, 4)$ to $(5, 0)$, so the second entry becomes zero.\n",
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