Several trace inequalities for positive semi-definite $2\times 2$ block matrices are revisited. We present shorter proofs of these inequalities along with their refinements. We obtain bounds for the smallest and the largest eigenvalues of a positive semi-definite $2\times 2$ block matrix involving the traces of its blocks.

中文翻译：

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涉及半正定分块矩阵的迹不等式

半正定的几个迹不等式$\mathrm{2个}\times \mathrm{2个}$重新访问块矩阵。我们提供了这些不等式的较短证明及其改进。我们获得半正定的最小和最大特征值的界限$\mathrm{2个}\times \mathrm{2个}$块矩阵涉及其块的痕迹。