Majorization
Definition
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.
Minorization
We will occasionally say that minorizes
, and write
, if
.
Alternative Criteria
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.
See Also
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