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equivalent condition to nilpotent

Source: 2016 South Korea USCM P8

August 17, 2020
linear algebraNilpotentmatrixcommutatorcollege contests

Problem Statement

For a n×nn\times n complex valued matrix AA, show that the following two conditions are equivalent. (i) There exists a n×nn\times n complex valued matrix BB such that AB−BA=AAB-BA=A. (ii) There exists a positive integer kk such that Ak=OA^k = O. (OO is the zero matrix.)
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