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A Trace Inequality for Positive Definite Matrices  
 
  Authors: Elena-Veronica Belmega, Samson Lasaulce, Mérouane Debbah,  
  Keywords: Trace inequality, positive definite matrices, positive semidefinite matrices.  
  Date Received: 09/11/08  
  Date Accepted: 24/01/09  
  Subject Codes:

15A45

  
  Editors: Fuzhen Zhang,  
 
  Abstract:

In this note we prove that $ mathrm{Tr}left{ mathbf{M} mathbf{N} + mathbf{P} mathbf{Q} ight} geq 0$ when the following two conditions are met: (i) the matrices $ mathbf{M}, mathbf{N}, mathbf{P}, mathbf{Q}$ are structured as follows $ mathbf{M} = mathbf{A} - mathbf{B}$, $ mathbf{N} = mathbf{B}^{-1} - mathbf{A}^{-1}$, $ mathbf{P} = mathbf{C} - mathbf{D}$, $ mathbf{Q} = (mathbf{B}+mathbf{D})^{-1} -(mathbf{A}+mathbf{C})^{-1}$ (ii) $ mathbf{A}$, $ mathbf{B}$ are positive definite matrices and $ mathbf{C}$, $ mathbf{D}$ are positive semidefinite matrices.;


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