A new proof of Lee's conjecture on the Frobenius norm via the matrix Cauchy-Schwarz inequality
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916841764421632 |
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| author | Zhang, Teng |
| author_facet | Zhang, Teng |
| contents | In 2010, Eun-Young Lee conjectured that if $A,B$ are two $n\times n$ complex matrices and $\left|A\right|, \left|B\right|$ are the absolute values of $A, B$, respectively, then \[ \|A+B\|_F\le \sqrt{\dfrac{1+\sqrt{2}}{2}}\|\left|A\right|+\left|B\right|\|_F, \] where $\|\cdot\|_F$ is the Frobenius norm of matrices. This conjecture has been proven by Lin and Zhang [J. Math. Anal. Appl. 516 (2022) 126542] by studying inequalities for the angle between two matrices induced by the Frobenius inner product. In this paper, we present a new proof of the same result, relying solely on the Cauchy-Schwarz inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02684 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new proof of Lee's conjecture on the Frobenius norm via the matrix Cauchy-Schwarz inequality Zhang, Teng Functional Analysis 15A60, 47A30 In 2010, Eun-Young Lee conjectured that if $A,B$ are two $n\times n$ complex matrices and $\left|A\right|, \left|B\right|$ are the absolute values of $A, B$, respectively, then \[ \|A+B\|_F\le \sqrt{\dfrac{1+\sqrt{2}}{2}}\|\left|A\right|+\left|B\right|\|_F, \] where $\|\cdot\|_F$ is the Frobenius norm of matrices. This conjecture has been proven by Lin and Zhang [J. Math. Anal. Appl. 516 (2022) 126542] by studying inequalities for the angle between two matrices induced by the Frobenius inner product. In this paper, we present a new proof of the same result, relying solely on the Cauchy-Schwarz inequality. |
| title | A new proof of Lee's conjecture on the Frobenius norm via the matrix Cauchy-Schwarz inequality |
| topic | Functional Analysis 15A60, 47A30 |
| url | https://arxiv.org/abs/2507.02684 |