We say a nonincreasing sequence of real numbers majorizes another nonincreasing sequence , and write if and only if all for all , , with equality when . If and are not necessarily nonincreasing, then we still write if this is true after the sequences have been sorted in nonincreasing order.
We will occasionally say that minorizes , and write , if .
It is also true that if and only if for all , , with equality when . An interesting consequence of this is that the finite sequence majorizes if and only if minorizes .
We can also say that this is the case if and only if for all ,
Both of these conditions are equivalent to our original definition.
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