Speech enhancement is important for applications such as telecommunications, hearing aids, automatic speech recognition and voice-controlled systems. Enhancement algorithms aim to reduce interfering noise and reverberation while minimizing any speech distortion. In this work for speech enhancement, we propose to use polynomial matrices to model the spatial, spectral and temporal correlations between the speech signals received by a microphone array and polynomial matrix eigenvalue decomposition (PEVD) to decorrelate in space, time and frequency simultaneously. We then propose a blind and unsupervised PEVD-based speech enhancement algorithm. Simulations and informal listening examples involving diverse reverberant and noisy environments have shown that our method can jointly suppress noise and reverberation, thereby achieving speech enhancement without introducing processing artefacts into the enhanced signal.