Linear mappings which are rank-k nonincreasing, II |
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Authors: | Raphael Loewy |
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Institution: |
a Department of Mathematics, Technion-Israel Institute of Technology, Haifa, Israel |
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Abstract: | We prove the following result. Let F be an infinite field of characteristic other than two. Let k be a positive integer. Let Sn(F) denote the space of all n × n symmetric matrices with entries in F, and let T:Sn(F)→Sn(F) be a linear operator. Suppose that T is rank-k nonincreasing and its image contains a matrix with rank higher than K. Then, there exist λεF and PεFn,n such that T(A)=λPAPt for all AεSn(F). λ can be chosen to be 1 if F is algebraically closed and ±1 if F=R, the real field. |
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