In order to improve the detection of malicious tampering of images, it is necessary to decrease the fragility of hidden watermarks, even for digital images which have been distorted incidentally. In this paper, we propose a new semi-fragile digital watermarking technique based on eigenvalues and eigenvectors of real symmetric matrix generated by the pair of four pixels. A signature bit for detecting malicious tampering of an image is generated using the dominant eigenvector. The dominant eigenvalue can reduce the sensitivity of quantization based watermarking. The experimental results show that this algorithm can resist high quality (70 or more) JPEG compression, and improve the detection performance of various malicious alterations.