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        "title": "Alfred Horn, Doubly Stochastic Matrices and the Diagonal of a Rotation Matrix (AJM76,1954,620–630; facsimile)",
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        "description": "Finite-dimensional statements with their real-vector and determinant conditions, not a classification of arbitrary infinite operators or all complex-vector majorization. Orthostochastic realizability of y from x does not mean every doubly stochastic matrix is orthostochastic; the original gives a counterexample and explicitly shows failure of the analogous complex-vector assertion. The majorization necessity is credited to Schur and the Hardy–Littlewood–Polya/Birkhoff inputs are inherited, not Horn discoveries.",
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      "statement": "# Horn realizes majorized diagonals and characterizes rotation-matrix diagonals\n\nFor finite real vectors x and y, majorization of y by x, including equality of total sums, admits an orthostochastic realization y_i=sum_j U_ij^2 x_j with a real orthogonal matrix U. Consequently these conditions exactly characterize diagonals of Hermitian matrices with prescribed real eigenvalues, with real symmetric realization available. Horn also proves that diagonals of real determinant+1 orthogonal matrices form the convex hull of sign vectors with an even number of negative coordinates, deriving orthogonal and unitary diagonal characterizations.\n\n## Scientific scope\n\nFinite-dimensional statements with their real-vector and determinant conditions, not a classification of arbitrary infinite operators or all complex-vector majorization. Orthostochastic realizability of y from x does not mean every doubly stochastic matrix is orthostochastic; the original gives a counterexample and explicitly shows failure of the analogous complex-vector assertion. The majorization necessity is credited to Schur and the Hardy–Littlewood–Polya/Birkhoff inputs are inherited, not Horn discoveries.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nAlfred Horn\n\nSource-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.\n\n\n\n## Original sources\n\n- [Alfred Horn, Doubly Stochastic Matrices and the Diagonal of a Rotation Matrix (AJM76,1954,620–630; facsimile)](https://www.convexoptimization.com/TOOLS/AHorn.pdf)\n",
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      "title": "Horn realizes majorized diagonals and characterizes rotation-matrix diagonals"
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    "id": "horn-majorized-diagonal-realization-rotation-1954",
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    "title": "Horn realizes majorized diagonals and characterizes rotation-matrix diagonals",
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    "what_changed": "For finite real vectors x and y, majorization of y by x, including equality of total sums, admits an orthostochastic realization y_i=sum_j U_ij^2 x_j with a real orthogonal matrix U. Consequently these conditions exactly characterize diagonals of Hermitian matrices with prescribed real eigenvalues, with real symmetric realization available. Horn also proves that diagonals of real determinant+1 orthogonal matrices form the convex hull of sign vectors with an even number of negative coordinates, deriving orthogonal and unitary diagonal characterizations.",
    "scope": "For finite real vectors x and y, majorization of y by x, including equality of total sums, admits an orthostochastic realization y_i=sum_j U_ij^2 x_j with a real orthogonal matrix U. Consequently these conditions exactly characterize diagonals of Hermitian matrices with prescribed real eigenvalues, with real symmetric realization available. Horn also proves that diagonals of real determinant+1 orthogonal matrices form the convex hull of sign vectors with an even number of negative coordinates, deriving orthogonal and unitary diagonal characterizations. Finite-dimensional statements with their real-vector and determinant conditions, not a classification of arbitrary infinite operators or all complex-vector majorization. Orthostochastic realizability of y from x does not mean every doubly stochastic matrix is orthostochastic; the original gives a counterexample and explicitly shows failure of the analogous complex-vector assertion. The majorization necessity is credited to Schur and the Hardy–Littlewood–Polya/Birkhoff inputs are inherited, not Horn discoveries.",
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