In this paper, we propose a novel privacy-preserving machine learning scheme with encrypted images, called EtC (Encryption-then-Compression) images. Using machine learning algorithms in cloud environments has been spreading in many fields. However, there are serious issues with it for end users, due to semi-trusted cloud providers. Accordingly, we propose using EtC images, which have been proposed for EtC systems with JPEG compression. In this paper, a novel property of EtC images is considered under the use of z-score normalization. It is demonstrated that the use of EtC images allows us not only to protect visual information of images, but also to preserve both the Euclidean distance and the inner product between vectors. In addition, dimensionality reduction is shown to can be applied to EtC images for fast and accurate matching. In an experiment, the proposed scheme is applied to a facial recognition algorithm with classifiers for confirming the effectiveness of the scheme under the use of support vector machine (SVM) with the kernel trick.