Symmetric eigenvalue decomposition (EVD) is a fundamental analytic and numerical tool used in many scientific areas. The state-of-the-art algorithm in terms of performance is typically the two-stage tridiagonalization method. The first stage in the two-stage tridiagonalization is called successive band reduction (SBR), which reduces a symmetric matrix to a band form, and its computational cost usually dominates. When Tensor Core (specialized matrix computational accelerator) is used to accelerate the expensive EVD, the conventional ZY-representation-based method results in suboptimal performance due to unfavorable shapes of the matrix computations. In this paper, we propose a new method that uses WY representation instead of ZY representation (see Section~\ref{WY} for details), which can provide a better combination of locality and parallelism so as to perform better on Tensor Cores. Experimentally, the proposed method can bring up to 3.7x speedup in SBR and 2.3x in the entire EVD compared to state-of-the-art implementations.