In this technical report, we analyze Legendre decomposition for non-negative tensor in theory and application. In theory, the properties of dual parameters and dually flat manifold in Legendre decomposition are reviewed, and the process of tensor projection and parameter updating is analyzed. In application, a series of verification experiments and clustering experiments with parameters on submanifold were carried out, hoping to find an effective lower dimensional representation of the input tensor. The experimental results show that the parameters on submanifold have no ability to be directly used as low-rank representations. Combined with analysis, we connect Legendre decomposition with neural networks and low-rank representation applications, and put forward some promising prospects.