What is recorded here?
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.
Source-linked catalogue note. A detailed explanatory guide has not yet been prepared for this entry.
Scientific scope
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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People and systems
Alfred Horn
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.
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English catalogue narrative
Horn realizes majorized diagonals and characterizes rotation-matrix diagonals
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
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.
Reading status
This 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.
People and systems
Alfred Horn
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.