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The Quaternion Matrix-Valued Young's Inequality  
 
  Authors: Renying Zeng,  
  Keywords: Quaternion, Matrix, Young's inequality, Real representation.  
  Date Received: 09/12/02  
  Date Accepted: 04/06/05  
  Subject Codes:

15A45, 15A42.

  
  Editors: George P. H. Styan,  
 
  Abstract:

In this paper, we prove Young's inequality in quaternion matrices: for any $ ntimes n$ quaternion matrices $ A$ and $ B$, any $ p,qin (1,infty )$ with $ frac{1}{p}+frac{1}{q}=1$, there exists $ ntimes n$ unitary quaternion matrix $ U$such that $ Uvert AB^{ast} vert U^{ast} leq tfrac{1}{p}vert Avert^{p}+tfrac{1}{q} vert Bvert^{q}.$

Furthermore, there exists unitary quaternion matrix $ U$ such that the equality holds if and only if $ vert Bvert=vert Avert^{p-1}$.;


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