majorization
For any real vector , let denote the components of in non-increasing order.
For , we say that is majorized by , or majorizes , if
A common notation for “ is majorized by ” is .
Remark:
A canonical example is that, if , are non-negative real numbers such that their sum is equal to 1, then
In general, vaguely means that the components of is less spread out than are the components of .
Reference
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G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, 2nd edition, 1952, Cambridge University Press, London.
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A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, 1979, Acadamic Press, New York.
Title | majorization |
---|---|
Canonical name | Majorization |
Date of creation | 2013-03-22 14:30:22 |
Last modified on | 2013-03-22 14:30:22 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 8 |
Author | rspuzio (6075) |
Entry type | Definition |
Classification | msc 26D99 |
Defines | majorize |
Defines | majorization |