This paper presents a random set based approach to tracking of an unknown number of extended targets, in the presence of clutter measurements and missed detections, where the targets' extensions are modeled as random matrices. For this purpose, the random matrix framework developed recently by Koch et al. is adapted into the extended target PHD framework, resulting in the Gaussian inverse Wishart PHD (GIW-PHD) filter. A suitable multiple target likelihood is derived, and the main filter recursion is presented along with the necessary assumptions and approximations. The particularly challenging case of close extended targets is addressed with practical measurement clustering algorithms. The capabilities and limitations of the resulting extended target tracking framework are illustrated both in simulations and in experiments based on laser scans.