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Let $N$ be an $n \times n$ matrix with coefficients in the field $F .$ The matrix $N$ is said to be nilpotent if some power of $N$ is the zero matrix, i.e., $N^{k}=0$ for some $k .$ Prove that any nilpotent matrix is similar to a block diagonal matrix whose blocks are matrices with 1 's along the first superdiagonal and 0 's elsewhere.

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