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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $v \\in \\mathbb{K}^n$ be a vector with $v^* v = 1$, where $v^*$ is the [[conjugate transpose|0EX045E2BP]] of $v$.\nThe *Householder transformation* (or *Householder reflector*) defined by $v$ is the $n \\times n$ matrix\n$$P = I - 2 v v^*.$$\n\n## Remarks\n\n$P$ is Hermitian and [[unitary|0D6RMJ10M9]], and $P^2 = I$; for $\\mathbb{K} = \\R$ it is a symmetric [[orthogonal matrix|0QRBVFJ50X]] with determinant $-1$.\nIt fixes every vector orthogonal to $v$ and sends $v$ to $-v$, so it is the [[reflection|025158VEYZ]] in the hyperplane $v^\\perp$.\nIf $x, y \\in \\R^n$ are distinct with $\\norm{x} = \\norm{y}$, then $v = (x - y)/\\norm{x - y}$ gives $P x = y$.\nExample: in $\\R^2$, take $x = (3, 4)$ and $y = (5, 0)$; then $v = (-2, 4)/\\sqrt{20}$, and\n$$P = I - \\frac{2}{20} \\begin{pmatrix} 4 \u0026 -8 \\\\ -8 \u0026 16 \\end{pmatrix} = \\frac{1}{5} \\begin{pmatrix} 3 \u0026 4 \\\\ 4 \u0026 -3 \\end{pmatrix},$$\nand indeed $P (3, 4)^T = (5, 0)^T$.\nNon-example: $v = (2, 0)$ is not a unit vector, and $I - 2 v v^T = \\begin{pmatrix} -7 \u0026 0 \\\\ 0 \u0026 1 \\end{pmatrix}$ is not orthogonal.\nFor a nonzero $u$ that is not normalized, one writes $P = I - 2 u u^* / (u^* u)$.\nIn computations, $P$ is never formed: $P A = A - 2 v (v^* A)$ costs $O(n m)$ operations for an $n \\times m$ matrix $A$.\n\n## Explanations\n\n### L1 human\n\nA Householder transformation is a mirror reflection written as a matrix.\nThe unit vector $v$ points straight out of the mirror.\nReflecting twice gives back the original vector, and lengths never change.\n",
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