Kunze Linear Algebra · Problem 7.3.10

Rational and Jordan Forms

Mathematics — Methods · nilpotent-square-root

Problem

Let nn be a positive integer, n≥2n \geq 2, and let NN be an n×nn \times n matrix over the field FF such that Nn=0N^n = 0 but Nn−1≠0N^{n-1} \neq 0. Prove that NN has no square root, i.e., that there is no n×nn \times n matrix AA such that A2=NA^2 = N.

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