Solution: 2010-10 Metric space of matrices

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

The best solution was submitted by Gee Won Suh (서기원), 2009학번. Congratulations!

Here is his Solution of Problem 2010-10.

Alternative solutions were submitted by 정성구 (수리과학과 2007학번, +3), 김치헌 (수리과학과 2006학번, +3), 강동엽 (2009학번, +2).

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1 thought on “Solution: 2010-10 Metric space of matrices

  1. diya

    excellent. i did it a different way. tried to use binomial expansion and then got stuck cause i dint know what to do with epsilon. this is much more elegant.

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