A **nilpotent matrix** is a square matrix \( A \) such that there exists some positive integer \( k \) for which the matrix raised to the power of \( k \) equals the zero matrix.
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A square matrix is nilpotent when one of its positive powers is zero. It represents a nilpotent linear map in a chosen basis; changing basis preserves nilpotence. Over an algebraically closed field, its Jordan normal form has only zero-eigenvalue blocks.