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Abstract
In this short note, we prove that given any nonsingular matrix, the singular values of the superoptimal preconditioned matrix do not exceed the corresponding singular values of the optimal preconditioned matrix.
Acknowledgements
The authors would like to thank the guest editor Professor Ke Chen and the referee for their detailed comments on the paper. The research of X.-Q. Jin is supported by the research grant 050/2005/A from FDCT. The research of Y.-M. Wei is supported by the National Natural Science Foundation of China under the grant 10471027 and Shanghai Education Committee.