DDVV-type inequality for Hermitian matrices

Ge, Jianquan; Xu, Song; You, Hangyu; Zhou, Yi

doi:10.1016/j.laa.2017.04.028

Cited by 3 publications

(5 citation statements)

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“…We have already known the DDVV-type inequality for (skew-)Hermitian matrices and its equality condition (cf. [18]). In this section, we firstly give a simpler proof of this result to illustrate our main technique of this paper.…”

Section: Ddvv-type Inequality For Complex Matricesmentioning

confidence: 99%

“…Ge [14] proved the DDVV-type inequality for real skew-symmetric matrices, and applied it to get a Simons-type inequality for Yang-Mills fields in Riemannian submersion geometry. Ge-Xu-You-Zhou [18] extended the DDVV-type inequalities from real symmetric and skew-symmetric matrices to Hermitian and skew-Hermitian matrices. In Section 2 of this paper, by using the DDVV-type inequalities for real symmetric and skew-symmetric matrices, we will firstly give a much simpler proof of the DDVV-type inequality for (skew-)Hermitian matrices.…”

Section: Introductionmentioning

confidence: 99%

“…The optimal constant c in this case is 4 3 when m ≥ 3. We summarize the DDVV-type inequalities mentioned above by indicating the optimal constant c with respect to the types of the matrices in the following Table 1 (see [15,14,18], Theorem 2.3 and Corollaries 2.5, 2.6 in Section 2 for equality conditions).…”

Section: Introductionmentioning

confidence: 99%

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Some generalizations of the DDVV and BW inequalities

Zhou

2018

Preprint

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In this paper we generalize the known DDVV-type inequalities for real (skew-)symmetric and complex (skew-)Hermitian matrices into arbitrary real, complex and quaternionic matrices. Inspired by the Erdős-Mordell inequality, we establish the DDVV-type inequalities for matrices in the subspaces spanned by a Clifford system or a Clifford algebra. We also generalize the Böttcher-Wenzel inequality to quaternionic matrices.

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Section: Ddvv-type Inequality For Complex Matricesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Some generalizations of the DDVV and BW inequalities

Zhou

2018

Preprint

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“…We remind that for general Hermitian matrices the optimal constant c = 4 3 is bigger than 1 here (cf. Section 1, [17], [18]).…”

Section: The Equality Holds If and Only If |Xmentioning

confidence: 99%

“…[14]); c = 4 3 for Hermitian matrices (cf. [17]) and also for arbitrary real or complex matrices (cf. [18]).…”

Section: Introductionmentioning

confidence: 99%

On some conjectures by Lu and Wenzel

et al. 2020

Linear Algebra and its Applications

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In order to give a unified generalization of the BW inequality and the DDVV inequality, Lu and Wenzel proposed three Conjectures 1, 2, 3 and an open Question 1 in 2016. In this paper we discuss further these conjectures and put forward several new conjectures which will be shown equivalent to Conjecture 2. In particular, we prove Conjecture 2 and hence all conjectures in some special cases. For Conjecture 3, we obtain a bigger upper bound 2 + √ 10/2, and we also give a weaker answer for the more general Question 1. In addition, we obtain some new simple proofs of the complex BW inequality and the condition for equality.2010 Mathematics Subject Classification. 15A45, 15B57, 53C42.

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