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In this paper, we study the adversarial robustness of deep neural networks (DNN) for classification against optimal classifiers. We look at the smallest magnitude of possible additive perturbations that can change a classifier's output. We provide a matrix-theoretic explanation of the adversarial fragility of DNNs for classification. In particular, our theoretical results show that the adversarial robustness of a neural network can degrade as the input dimension d increases. Analytically, we show that the adversarial robustness of neural networks can be only 1/√d of the best possible adversarial robustness of optimal classifiers. Our theories match remarkably well with empirical results. The matrix-theoretic explanation aligns with an earlier information-theoretic feature-compression-based explanation for the adversarial fragility of neural networks.