Tag Archives: nilpotent

2013-14 Nilpotent matrix

Let \(A, B\) are \(N \times N \) complex matrices satisfying \( rank(AB – BA) = 1 \). Prove that \( (AB – BA)^2 = 0 \).

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Let M_n×n be the space of real n×n matrices, regarded as a metric space with the distance function

\(\displaystyle d(A,B)=\sum_{i,j} |a_{ij}-b_{ij}|\)

for A=(a_ij) and B=(b_ij).
Prove that \(\{A\in M_{n\times n}: A^m=0 \text{ for some positive integer }m\}\) is a closed set.

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