The ubiquity of location-based devices has facilitated massive trajectory data collection, while such data holds immense value, it has the risk of privacy leakage. Local differential privacy (LDP) has emerged as a promising privacy-preserving method for trajectory synthesis, offering formal privacy guarantees while supporting location-based services. However, the noise introduced by LDP significantly degrades the utility of synthetic trajectory data, leading to several key challenges: excessive noise, low utility, and poor data fidelity. To enhance synthetic data quality, multiple metrics need to be considered, such as mobility model, trajectory length, and density. Yet directly estimating all these metrics under LDP constraints would introduce substantial error. To address this, we propose an LDP trajectory synthesis framework that strategically employs singular value decomposition (SVD) and low-rank approximation. Our approach offers two main advantages: (1) SVD reduces the mobility model dimensionality, thereby decreasing noise requirements. (2) Low-rank approximation focuses privacy protection on dominant movement patterns while minimizing overall noise impact. This method eliminates the need to upload raw trajectories, reduces privacy risks, and generates synthetic data that more accurately preserves real-world movement characteristics - making it more useful for downstream applications. Experimental results on real-world datasets confirm that our approach maintains strong privacy guarantees while improving both data quality and computational efficiency.