SWhat changed
Schur proves that every finite square complex matrix is unitarily similar to a triangular matrix whose diagonal lists its eigenvalues with algebraic multiplicity. He derives sum |lambda_j|^2 <= sum |a_ij|^2, with equality exactly when a unitary change of basis diagonalizes the matrix, and related bounds on the real and imaginary parts of the eigenvalues.
Scientific scope
Schur proves that every finite square complex matrix is unitarily similar to a triangular matrix whose diagonal lists its eigenvalues with algebraic multiplicity. He derives sum |lambda_j|^2 <= sum |a_ij|^2, with equality exactly when a unitary change of basis diagonalizes the matrix, and related bounds on the real and imaginary parts of the eigenvalues. The original displays lower triangular form; transposition of convention gives the usual upper triangular statement. No distinct-eigenvalue or diagonalizability hypothesis is imposed for triangularization. Real orthogonal triangularization is asserted only when both the matrix and all its eigenvalues are real. Equality characterizes unitary diagonalizability, not arbitrary similarity to a diagonal matrix. This record does not claim an infinite-dimensional triangularization theorem, a numerical Schur algorithm, or the paper's later integral-equation results. No infinite-dimensional theorem or numerical algorithm. A real matrix with nonreal eigenvalues need not have a real orthogonal triangularization. Autonne/Stickelberger precedents are cited. Only original PartI pp488–496 selected, not later integral-equation PartII.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The original displays lower triangular form; transposition of convention gives the usual upper triangular statement. No distinct-eigenvalue or diagonalizability hypothesis is imposed for triangularization. Real orthogonal triangularization is asserted only when both the matrix and all its eigenvalues are real. Equality characterizes unitary diagonalizability, not arbitrary similarity to a diagonal matrix. This record does not claim an infinite-dimensional triangularization theorem, a numerical Schur algorithm, or the paper's later integral-equation results. No infinite-dimensional theorem or numerical algorithm. A real matrix with nonreal eigenvalues need not have a real orthogonal triangularization. Autonne/Stickelberger precedents are cited. Only original PartI pp488–496 selected, not later integral-equation PartII.
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 2026-10-23.5. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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