AWhat 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.
Scientific 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.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. 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.
Contributors
- The named authors in the research record · SCIENCE: Source-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.
Provisional WRS assessment, edition 2026-10-23.3. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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