The ability to obtain accurate estimates of city-wide urban traffic patterns is essential for the development of effective intelligent transportation systems and the efficient operation of smart mobility platforms. This paper focuses on the network-wide traffic speed estimation, using trajectory data generated by a city-wide fleet of ride-sourcing vehicles equipped with GPS-capable smartphones. A cell-based map-matching technique is proposed to link vehicle trajectories with road geometries, and to produce network-wide spatio-temporal speed matrices. Data limitations are addressed using the Schatten p-norm matrix completion algorithm, which can minimize speed estimation errors even with high rates of data unavailability. A case study using data from Chengdu, China, demonstrates that the algorithm performs well even in situations involving continuous data loss over a few hours, and consequently, addresses large-scale network-wide traffic state estimation problems with missing data, while at the same time outperforming other data recovery techniques that were used as benchmarks. Our approach can be used to generate congestion maps that can help monitor and visualize traffic dynamics across the network, and therefore form the basis for new traffic management, proactive congestion identification, and congestion mitigation strategies.