In 2005, B¨ottcher and Wenzel proposed a conjecture that kXY − Y XkF ≤ √ 2kXkF kY kF for any X, Y ∈ C n×n , where k · kF denotes the Frobenius norm. Then they proved it for the case of 2 × 2 matrices. Later, L´aszl´o proved the result for the case of 3 × 3 and Vong and Jin proved the conjecture for general n × n real matrices. In this thesis, we consider a norm inequality of the commutator for even-order tensors, which is a higher-order generalization of B¨ottcher-Wenzel conjecture. Firstly, we give an introduction of the research background in Chapter 1. In Chapter 2, some preliminaries of tensors are given. Then we introduce the multiplication, transpose and conjugate transpose for tensors. Furthermore, the Frobenius norm of tensors and the concept of Hermitian tensor are proposed. They are the precondition of the proposition. In Chapter 3, we embark on the process of the proof of this proposition. An equivalent transformation of the proposition is raised. We prove its equivalent form instead of the original problem. At last, we give a brief conclusion for the thesis. Some possible furture works on this tensor version B¨ottcher-Wenzel conjecture are proposed.