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        "description": "The admitted algorithm is real orthogonal-reflection triangularization, particularly for nonsingular inversion and full-column-rank least squares. Zero/aligned columns need lawful skip/identity branches. The original arbitrary-complex lemma omits a needed phase condition and is not adopted here; later phase-aware repair is separate. Exact norm preservation is not blanket floating-point stability or a universal ill-conditioned pseudoinverse solution.",
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      "statement": "# Householder: triangularizing real matrices with orthogonal reflections\n\nHouseholder uses successive orthogonal reflections to triangularize a real matrix while preserving Euclidean geometry, reducing the square-root count compared with a sequence of plane rotations.\n\nA large system of linear equations becomes easier to solve when its matrix is triangular. Householder clears a whole part of one column with a reflection, then repeats on the remaining block. Orthogonal transformations preserve lengths, so the reduction avoids changing the matrix's exact spectral conditioning. The original paper also connects the result to rectangular least squares and analyzes the arithmetic saved relative to plane rotations. These advantages do not by themselves certify every floating-point implementation. This record retains the real method; the original's arbitrary-complex phase assertion needs correction and is not treated as established.\n\n## Scientific scope\n\nThe admitted algorithm is real orthogonal-reflection triangularization, particularly for nonsingular inversion and full-column-rank least squares. Zero/aligned columns need lawful skip/identity branches. The original arbitrary-complex lemma omits a needed phase condition and is not adopted here; later phase-aware repair is separate. Exact norm preservation is not blanket floating-point stability or a universal ill-conditioned pseudoinverse solution.\n\n## People and systems\n\nAlston S. Householder\n\n\n\n## Original sources\n\n- [Alston S. Householder, Unitary Triangularization of a Nonsymmetric Matrix (1958), original article hosted by Stanford](https://web.stanford.edu/class/cme324/classics/householder.pdf)\n- [MIT Journal of the ACM bibliography: David D. Morrison, Remarks on the Unitary Triangularization of a Nonsymmetric Matrix (1960)](https://projects.csail.mit.edu/jacm/References/morrison1960%3A185.html)\n- [Stanford-hosted Trefethen classic numerical-paper list and1993course account](https://web.stanford.edu/class/cme324/classics/)\n",
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        "url": "https://web.stanford.edu/class/cme324/classics/householder.pdf",
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