Autoencoders (AE) aim to reproduce the output from the input. They may hence tend to overfit towards learning the identity-function between the input and output, i.e., they may predict each feature in the output from itself in the input. This is not useful, however, when AEs are used for prediction tasks in the presence of noise in the data. It may seem intuitively evident that this kind of overfitting is prevented by training a denoising AE, as the dropped-out features have to be predicted from the other features. In this paper, we consider linear autoencoders, as they facilitate analytic solutions, and first show that denoising / dropout actually prevents the overfitting towards the identity-function only to the degree that it is penalized by the induced L2-norm regularization. In the main theorem of this paper, we show that the emphasized denoising AE is indeed capable of completely eliminating the overfitting towards the identity-function. Our derivations reveal several new insights, including the closed-form solution of the full-rank model, as well as a new (near-)orthogonality constraint in the low-rank model. While this constraint is conceptually very different from the regularizers recently proposed, their resulting effects on the learned embeddings are empirically similar. Our experiments on three well-known data-sets corroborate the various theoretical insights derived in this paper.