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Singular Values of Sums of Positive Semidefinite Matrices

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For positive real numbers a,b,a + b ≤ max {a + b1/2 a1/2,b+a1/2b1/2}.In this note,we generalize this fact to matrices by proving that for positive semidefinite matrices A and B of order n,for any c∈[-1,1]and j=1,2,…,n,sj(A+B)≤sj((A ⊕ B) + φc(A,B)) ≤ sj(A+ | B1/2A1/2 |) ⊕ (B+| A1/2B1/2 |),where sj(X) denotes the j-th largest singular value of X and φc(A,B):=1/2((1+c)|B1/2A1/2| (1-c)A1/2B1/2 (1-c)B1/2A1/2 (1+c)|A1/2B1/2|).This result sharpens some known result.Meanwhile,some related results are established.

singular valuespositive semidefinite matricesmajorizationunitarily invariant norms

CHEN Dongjun、ZHANG Yun

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School of Mathematical Sciences, Huaibei Normal University,Huaibei 235000, Anhui, China

Supported by the Natural Science Foundation of Anhui ProvinceNatural Science Foundation of Anhui Higher Education Institutions of ChinaNatural Science Foundation of Anhui Higher Education Institutions of China

1708085QA05KJ2019A0588KJ2020ZD008

2020

10.19823/j.cnki.1007-1202.2020.0035

武汉大学自然科学学报(英文版)
武汉大学

武汉大学自然科学学报(英文版)

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影响因子：0.066
ISSN：1007-1202
年,卷(期)：2020.25(4)
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