We characterize bijective linear maps on $M_n(ℂ)$ that preserve the square roots of an idempotent matrix (of any rank). Every such map can be presented as a direct sum of a map preserving involutions and a map preserving square-zero matrices. Next, we consider bijective linear maps that preserve the square roots of a rank-one nilpotent matrix. These maps do not have standard forms when compared to similar linear preserver problems.

中文翻译：

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在保留幂等和一阶幂零矩阵平方根的地图上

我们在${米}_n(ℂ)$保留幂等矩阵（任意秩）的平方根。每个这样的映射都可以表示为保留对合的映射和保留零平方矩阵的映射的直接和。接下来，我们考虑保留一阶幂零矩阵的平方根的双射线性映射。与类似的线性保留问题相比，这些地图没有标准形式。