We show that any probability measure satisfying a Matrix Poincar\'e inequality with respect to some reversible Markov generator satisfies an exponential matrix concentration inequality depending on the associated matrix carr\'e du champ operator. This extends to the matrix setting a classical phenomenon in the scalar case. Moreover, the proof gives rise to new matrix trace inequalities which could be of independent interest. We then apply this general fact by establishing matrix Poincar\'{e} inequalities to derive matrix concentration inequalities for Gaussian measures, product measures and for Strong Rayleigh measures. The latter represents the first instance of matrix concentration for general matrix functions of negatively dependent random variables.

中文翻译：

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矩阵庞加莱不等式和集中

我们表明，关于某些可逆马尔可夫生成器，满足矩阵 Poincar\'e 不等式的任何概率度量都满足依赖于相关矩阵 carr\'e du champ 算子的指数矩阵浓度不等式。这扩展到矩阵设置在标量情况下的经典现象。此外，该证明产生了可能具有独立意义的新矩阵迹不等式。然后，我们通过建立矩阵 Poincar\'{e} 不等式来应用这个一般事实，以推导出高斯测度、乘积测度和强瑞利测度的矩阵浓度不等式。后者代表负相关随机变量的一般矩阵函数的矩阵集中的第一个实例。