Tridiagonalization, which is a key step in symmetric eigenvalue decomposition (EVD), aims to convert a symmetric matrix to a tridiagonal form. In Nvidia's cuSOLVER library, the FP64 precision tridiagonalization process only reach 2.1 TFLOPs out of 67 TFLOPs on H100 GPU, and it consumes a significant portion of the elapsed time in the entire EVD process, accounting for over 97%. Thus, improving the tridiagonalization performance is crucial on accelerating EVD. In this paper, we analyze the reasons behind the suboptimal performance of tridiagonalization on GPU architectures, and we propose a new double blocking band reduction algorithm along with an implementation of GPU-based bulge chasing to improve the tridiagonalization performance. Through experimental evaluation, the proposed FP64 precision tridiagonalization method yields up to 19.6 TFLOPs which is 9.3x and 5.2x faster compared cuSOVLER and MAGMA, respectively.