Majorization

A finite sequence of real numbers $\displaystyle A=a_1,a_2,\cdots,a_n$ is said to majorize a sequence $\displaystyle B=b_1,b_2,\cdots,b_n$ if and only if all of the following are true:

$\displaystyle a_1\geq b_1$

$\displaystyle a_1+a_2\geq b_1+b_2$

$\displaystyle \vdots$

$\displaystyle a_1+a_2+\cdots+a_{n-1}\geq b_1+b_2+\cdots+b_{n-1}$

$\displaystyle a_1+a_2+\cdots+a_n=b_1+b_2+\cdots+b_n$

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Majorization