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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2501.10603 |
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| _version_ | 1866929681784111104 |
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| author | Chang, Shih-Yu |
| author_facet | Chang, Shih-Yu |
| contents | Matrix inequalities play a pivotal role in mathematics, generalizing scalar inequalities and providing insights into linear operator structures. However, the widely used Löwner ordering, which relies on real-valued eigenvalues, is limited to Hermitian matrices, restricting its applicability to non-Hermitian systems increasingly relevant in fields like non-Hermitian physics. To overcome this, we develop a total ordering relation for complex numbers, enabling comparisons of the spectral components of general matrices with complex eigenvalues. Building on this, we introduce the Spectral and Nilpotent Ordering (SNO), a partial order for arbitrary matrices of the same dimensions. We further establish a theoretical framework for majorization ordering with complex-valued functions, which aids in refining SNO and analyzing spectral components. An additional result is the extension of the Schur--Ostrowski criterion to the complex domain. Moreover, we characterize Jordan blocks of matrix functions using a generalized dominance order for nilpotent components, facilitating systematic analysis of non-diagonalizable matrices. Finally, we derive monotonicity and convexity conditions for functions under the SNO framework, laying a new mathematical foundation for advancing matrix analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10603 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Matrix Ordering through Spectral and Nilpotent Structures in Totally Ordered Complex Number Fields Chang, Shih-Yu Functional Analysis Operator Algebras Quantum Physics Matrix inequalities play a pivotal role in mathematics, generalizing scalar inequalities and providing insights into linear operator structures. However, the widely used Löwner ordering, which relies on real-valued eigenvalues, is limited to Hermitian matrices, restricting its applicability to non-Hermitian systems increasingly relevant in fields like non-Hermitian physics. To overcome this, we develop a total ordering relation for complex numbers, enabling comparisons of the spectral components of general matrices with complex eigenvalues. Building on this, we introduce the Spectral and Nilpotent Ordering (SNO), a partial order for arbitrary matrices of the same dimensions. We further establish a theoretical framework for majorization ordering with complex-valued functions, which aids in refining SNO and analyzing spectral components. An additional result is the extension of the Schur--Ostrowski criterion to the complex domain. Moreover, we characterize Jordan blocks of matrix functions using a generalized dominance order for nilpotent components, facilitating systematic analysis of non-diagonalizable matrices. Finally, we derive monotonicity and convexity conditions for functions under the SNO framework, laying a new mathematical foundation for advancing matrix analysis. |
| title | Matrix Ordering through Spectral and Nilpotent Structures in Totally Ordered Complex Number Fields |
| topic | Functional Analysis Operator Algebras Quantum Physics |
| url | https://arxiv.org/abs/2501.10603 |