Symmetric　Positive　semi-Definite　(SPD)　matrices,　as　a　kind　of　effective　feature　descriptors,　have　been　widely　used　in　pattern　recognition　and　computer　vision　tasks.　Affine-invariant　metric　(AIM)　is　a　popular　way　to　measure　the　distance　between　SPD　matrices,　but　it　imposes　a　high　computational　burden　in　practice.　Compared　with　AIM,　the　Log-Euclidean　metric　embeds　the　SPD　manifold　via　the　matrix　logarithm　into　a　Euclidean　space　in　which　only　classical　Euclidean　computation　is　involved.　The　advantage　of　using　this　metric　for　the　non-linear　SPD　matrices　representation　of　data　has　been　recognized　in　some　domains　such　as　compressed　sensing,　however　one　pays　little　attention　to　this　metric　in　data　clustering.　In　this　paper,　we　propose　a　novel　Low　Rank　Representation　(LRR)　model　on　SPD　matrices　space　with　Log-Euclidean　metric　(LogELRR),　which　enables　us　to　handle　non-linear　data　through　a　linear　manipulation　manner.　To　further　explore　the　intrinsic　geometry　distance　between　SPD　matrices,　we　embed　the　SPD　matrices　into　Reproductive　Kernel　Hilbert　Space　(RKHS)　to　form　a　family　of　kernels　on　SPD　matrices　based　on　the　Log-Euclidean　metric　and　construct　a　novel　kernelized　LogELRR　method.　The　clustering　results　on　a　wide　range　of　datasets,　including　object　images,　facial　images,　3D　objects,　texture　images　and　medical　images,　show　that　our　proposed　methods　overcome　other　conventional　clustering　methods.　(C)　2017　Published　by　Elsevier　Ltd.