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Motivated by brain connectivity analysis and many other network data applications, we study the problem of estimating covariance and precision matrices and their differences across multiple populations. We propose a common reducing subspace model that leads to substantial dimension reduction and efficient parameter estimation. We explicitly quantify the efficiency gain through an asymptotic analysis. Our method is built upon and further extends a nascent technique, the envelope model, which adopts a generalized sparsity principle. This distinguishes our proposal from most xisting covariance and precision estimation methods that assume element-wise sparsity. Moreover, unlike most existing solutions, our method can naturally handle both covariance and precision matrices in a unified way, and work with matrix-valued data. We demonstrate the efficacy of our method through intensive simulations, and illustrate the method with an autism spectrum disorder data analysis.