Norm inequalities related to the arithmetic–geometric mean inequalities for positive semidefinite matrices

Positivity ◽  
2018 ◽  
Vol 22 (5) ◽  
pp. 1311-1324
Author(s):  
Mostafa Hayajneh ◽  
Saja Hayajneh ◽  
Fuad Kittaneh
Keyword(s):  
Geometric Mean ◽  
Positive Semidefinite ◽  
Positive Semidefinite Matrices ◽  
Norm Inequalities

scholarly journals A Geometric Mean of Parameterized Arithmetic and Harmonic Means of Convex Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Sangho Kum ◽  
Yongdo Lim
Keyword(s):  
Convex Functions ◽  
Convex Analysis ◽  
Geometric Mean ◽  
Positive Semidefinite ◽  
Dual Operator ◽  
Positive Semidefinite Matrices ◽  
Essential Properties ◽  
Harmonic Means ◽  
Further Development ◽  
Interesting Generalization

The notion of the geometric mean of two positive reals is extended by Ando (1978) to the case of positive semidefinite matricesAandB. Moreover, an interesting generalization of the geometric meanA # BofAandBto convex functions was introduced by Atteia and Raïssouli (2001) with a different viewpoint of convex analysis. The present work aims at providing a further development of the geometric mean of convex functions due to Atteia and Raïssouli (2001). A new algorithmic self-dual operator for convex functions named “the geometric mean of parameterized arithmetic and harmonic means of convex functions” is proposed, and its essential properties are investigated.

Norm inequalities for positive semidefinite matrices and a question of Bourin

2017 ◽  
Vol 28 (14) ◽  
pp. 1750102 ◽  
Author(s):  
Mostafa Hayajneh ◽  
Saja Hayajneh ◽  
Fuad Kittaneh
Keyword(s):  
Positive Semidefinite ◽  
Affirmative Answer ◽  
Unitarily Invariant Norm ◽  
Trace Norm ◽  
Positive Semidefinite Matrices ◽  
Norm Inequalities ◽  
Invariant Norm

Let [Formula: see text] such that [Formula: see text] and [Formula: see text] are positive semidefinite. It is shown that [Formula: see text] for [Formula: see text] and for every unitarily invariant norm. This gives an affirmative answer to one of the questions posed by Bourin regarding subadditivity inequalities in the case of the trace norm. New norm inequalities related to Bourin's question are also presented.

scholarly journals A trace bound for integer-diagonal positive semidefinite matrices

2020 ◽  
Vol 8 (1) ◽  
pp. 14-16
Author(s):  
Lon Mitchell
Keyword(s):  
Positive Semidefinite ◽  
Positive Semidefinite Matrix ◽  
Positive Semidefinite Matrices ◽  
Semidefinite Matrix

AbstractWe prove that an n-by-n complex positive semidefinite matrix of rank r whose graph is connected, whose diagonal entries are integers, and whose non-zero off-diagonal entries have modulus at least one, has trace at least n + r − 1.

scholarly journals Fischer Type Log-Majorization of Singular Values on Partitioned Positive Semidefinite Matrices

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Benju Wang ◽  
Yun Zhang
Keyword(s):  
Singular Values ◽  
Positive Semidefinite ◽  
Positive Semidefinite Matrices ◽  
Fischer’S Inequality ◽  
New Inequalities

In this paper, we establish a Fischer type log-majorization of singular values on partitioned positive semidefinite matrices, which generalizes the classical Fischer's inequality. Meanwhile, some related and new inequalities are also obtained.

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