We say that a square real matrix $M$ is \emph{off-diagonal nonnegative} if and only if all entries outside its diagonal are nonnegative real numbers. In this note we show that for any off-diagonal nonnegative symmetric matrix $M$, there exists a nonnegative symmetric matrix $\widehat{M}$ which is sparse and close in spectrum to $M$.

中文翻译：

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对称和非对角非负实矩阵的谱稀疏化边界

我们说方实矩阵 $M$ 是 \emph{off-diagonal nonnegative} 当且仅当其对角线以外的所有条目都是非负实数。在这篇笔记中，我们展示了对于任何非对角非负对称矩阵 $M$，都存在一个非负对称矩阵 $\widehat{M}$，它是稀疏的并且在频谱上接近 $M$。