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  Volume 9, Issue 1, Article 4
On Rank Subtractivity Between Normal Matrices

    Authors: Jorma K. Merikoski, Xiaoji Liu,  
    Keywords: Rank subtractivity, Minus partial ordering, Star partial ordering, Sharp partial ordering, Normal matrices, EP matrices.  
    Date Received: 13/07/2007  
    Date Accepted: 05/02/2008  
    Subject Codes:

15A45, 15A18.

    Editors: Fuzhen Zhang,  

The rank subtractivity partial ordering is defined on $ mathbb{C}^{ntimes n}$ ($ ngeq 2$) by $ mathbf{A}leq^-mathbf{B} Leftrightarrow {mathrm{rank}}(mathbf{B}-mathbf{A}) = {mathrm{rank}}mathbf{B}-{mathrm{rank}}mathbf{A}$, and the star partial ordering by $ mathbf{A}le^*mathbf{B}Leftrightarrow mathbf{A}^*mathbf{A} = mathbf{A}^*mathbf{B} mathrel{land}mathbf{A}mathbf{A}^* = mathbf{B}mathbf{A}^*$. If $ mathbf{A}$ and $ mathbf{B}$ are normal, we characterize $ mathbf{A}leq^-mathbf{B}$. We also show that then $ mathbf{A}leq^-mathbf{B}mathrel{land}mathbf{AB}= mathbf{BA} Leftrightar... ...arrow mathbf{A} leq^-mathbf{B} mathrel{land}mathbf{A}^2leq^-mathbf{B}^2$. Finally, we remark that some of our results follow from well-known results on EP matrices.

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