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Nilpotent Matrices

Source: IMC 2000 Day 2 Problem 6

October 27, 2020
linear algebramatrixalgebrapolynomial

Problem Statement

Let AA be a real n×nn\times n Matrix and define eA=∑k=0∞Akk!e^{A}=\sum_{k=0}^{\infty} \frac{A^{k}}{k!} Prove or disprove that for any real polynomial P(x)P(x) and any real matrices A,BA,B, P(eAB)P(e^{AB}) is nilpotent if and only if P(eBA)P(e^{BA}) is nilpotent.
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