Let \(M\) and \(N\) be \(n\times n\) matrices over \(\mathbb {C}\) with \(MN-NM=M\). Prove that \(M^kN-NM^k=kM^k\) for each \(k\ge 1\). Prove that \(M\) is nilpotent.
Question
Math 6310 • Algebra
Spring 2019 • Question 6
Question
Spring 2019 • Question 6
Let \(M\) and \(N\) be \(n\times n\) matrices over \(\mathbb {C}\) with \(MN-NM=M\). Prove that \(M^kN-NM^k=kM^k\) for each \(k\ge 1\). Prove that \(M\) is nilpotent.