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    "title": "Schur proves that entrywise products preserve positive semidefiniteness",
    "summary": "Schur proves that the coefficientwise product of two finite nonnegative Hermitian forms is again nonnegative. The proof factors one coefficient matrix into Gram terms and expresses the product form as a sum of nonnegative forms. He also bounds the product's characteristic roots using the extremal eigenvalues of one form and diagonal entries of the other, and derives a product-closure result for continuous nonnegative Hermitian kernels.",
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        "title": "Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen (JReineAngewMath140,1–28; selectedoriginalscans)",
        "url": "https://gdz.sub.uni-goettingen.de/download/pdf/PPN243919689_0140/LOG_0004.pdf",
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      "scope": {
        "description": "The product is entrywise in a common index basis, not ordinary matrix multiplication. Positive semidefinite and strictly positive definite are different: if one factor is positive definite and the other is positive semidefinite with strictly positive diagonal, the product is positive definite; zero rows/diagonal can prevent strictness. The stated kernel application retains the original continuity and finite-interval setting. The finite matrix statement and its continuous-kernel extension retain their distinct hypotheses; they do not assert every later theorem about infinite-dimensional operators or completely positive maps.",
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          "citation": "Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen (JReineAngewMath140,1–28; selectedoriginalscans)",
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      "statement": "# Schur proves that entrywise products preserve positive semidefiniteness\n\nSchur proves that the coefficientwise product of two finite nonnegative Hermitian forms is again nonnegative. The proof factors one coefficient matrix into Gram terms and expresses the product form as a sum of nonnegative forms. He also bounds the product's characteristic roots using the extremal eigenvalues of one form and diagonal entries of the other, and derives a product-closure result for continuous nonnegative Hermitian kernels.\n\n## Scientific scope\n\nThe product is entrywise in a common index basis, not ordinary matrix multiplication. Positive semidefinite and strictly positive definite are different: if one factor is positive definite and the other is positive semidefinite with strictly positive diagonal, the product is positive definite; zero rows/diagonal can prevent strictness. The stated kernel application retains the original continuity and finite-interval setting. The finite matrix statement and its continuous-kernel extension retain their distinct hypotheses; they do not assert every later theorem about infinite-dimensional operators or completely positive maps.\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\nJ. Schur\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- [Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen (JReineAngewMath140,1–28; selectedoriginalscans)](https://gdz.sub.uni-goettingen.de/download/pdf/PPN243919689_0140/LOG_0004.pdf)\n",
      "summary": "Schur proves that the coefficientwise product of two finite nonnegative Hermitian forms is again nonnegative. The proof factors one coefficient matrix into Gram terms and expresses the product form as a sum of nonnegative forms. He also bounds the product's characteristic roots using the extremal eigenvalues of one form and diagonal entries of the other, and derives a product-closure result for continuous nonnegative Hermitian kernels.",
      "title": "Schur proves that entrywise products preserve positive semidefiniteness"
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    "id": "schur-entrywise-product-positive-hermitian-forms-1911",
    "entry_id": "schur-entrywise-product-positive-hermitian-forms-1911",
    "revision": 1,
    "title": "Schur proves that entrywise products preserve positive semidefiniteness",
    "context": "Matrix analysis and positive kernels",
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    "what_changed": "Schur proves that the coefficientwise product of two finite nonnegative Hermitian forms is again nonnegative. The proof factors one coefficient matrix into Gram terms and expresses the product form as a sum of nonnegative forms. He also bounds the product's characteristic roots using the extremal eigenvalues of one form and diagonal entries of the other, and derives a product-closure result for continuous nonnegative Hermitian kernels.",
    "scope": "Schur proves that the coefficientwise product of two finite nonnegative Hermitian forms is again nonnegative. The proof factors one coefficient matrix into Gram terms and expresses the product form as a sum of nonnegative forms. He also bounds the product's characteristic roots using the extremal eigenvalues of one form and diagonal entries of the other, and derives a product-closure result for continuous nonnegative Hermitian kernels. The product is entrywise in a common index basis, not ordinary matrix multiplication. Positive semidefinite and strictly positive definite are different: if one factor is positive definite and the other is positive semidefinite with strictly positive diagonal, the product is positive definite; zero rows/diagonal can prevent strictness. The stated kernel application retains the original continuity and finite-interval setting. The finite matrix statement and its continuous-kernel extension retain their distinct hypotheses; they do not assert every later theorem about infinite-dimensional operators or completely positive maps.",
    "evidence_summary": "The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The product is entrywise in a common index basis, not ordinary matrix multiplication. Positive semidefinite and strictly positive definite are different: if one factor is positive definite and the other is positive semidefinite with strictly positive diagonal, the product is positive definite; zero rows/diagonal can prevent strictness. The stated kernel application retains the original continuity and finite-interval setting. The finite matrix statement and its continuous-kernel extension retain their distinct hypotheses; they do not assert every later theorem about infinite-dimensional operators or completely positive maps.",
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