Regular maintenance of civil engineering structures is essential for their safety. Current maintenance regimes involve periodic inspections at regular time intervals. In the time between inspections, there can be a critical development in the structural integrity of a structure, which can be expensive to repair or could even lead to structural failure. A more robust maintenance approach would involve continuous monitoring of the structure. This is the research area of Structural Health Monitoring (SHM). Vibration-based monitoring, which is a subset of SHM, aims to provide new cost-effective maintenance solutions that provide long-term life-safety benefits. Vibration-based monitoring uses vibration measurements from sensors to assess the "Health" of a structure. Damage in a structure will alter the structure's stiffness, mass and damping characteristics, which in turn will change the dynamic properties of the system. This change can then be discovered in the vibration data. The growth of the field is partly due to the significant advances in data-driven science and engineering in recent decades. One of the issues in vibration-based monitoring is the presence of operational and environmental variability in the vibration data. With this variability present, it is challenging to determine from vibration data the characteristics of the underlying dynamic system. The aim of this thesis is to use Robust Principal Component Analysis (rPCA) to reduce or eliminate the operational variability from the traffic to allow for environmental and damage detection. rPCA is a matrix factorisation method that decomposes a data matrix into a low-rank matrix L and a sparse matrix S. The reconstructed low-rank matrix L contains the main correlations in the data that are robust to outliers and corrupt data that are contained in the sparse matrix S. By applying rPCA to the frequency representation of the vibration data, it is hoped that the underlying coherent structure corresponding to the dynamic system can be recovered. The vibration data used for this thesis is from two measurement campaigns conducted on the Haringvlietbrug. The Haringvlietbrug is a steel box girder bridge in the Netherlands, and there are several fatigue cracks present in the bridge. This presented an opportunity for damage detection. The goal of the first measurement campaign is to conduct damage detection and discover if there is a difference between vibration data from a damaged area with fatigue cracks and a "healthy" reference area. In the second measurement campaign, the goal was to extract the underlying dynamics of the structure at different temperatures and see if it was possible to distinguish between the different structural states at different temperatures. After applying the rPCA on the vibration data, (regular) principal component analysis (PCA) is used to embed the data into the low-rank subspace of the PCs to distinguish between the different structural states. The rPCA was successful in extracting the coherent structures in the vibration data corresponding to the underlying dynamic properties of the system. In the subsequent PCA, vibration data with underlying different structural states had different scores in the first three PCs. In other words, it was possible to distinguish between the different structural states based on the first three PCs, which correspond to the main correlation within the data. For the first measurement campaign, this meant it was possible to distinguish between vibration data in the damaged area and the "healthy" reference area and detect "damage". However, there was a difference in the structural configuration between the two areas, so it was not possible to conclude that the differences in vibration data were due to damage caused by the fatigue cracks. In the second measurement campaign, the dynamic system properties at different temperatures were recovered. With the low-rank vibration data from the rPCA, it was possible to distinguish between vibration data with a 1°C difference in the first three PCs from the (regular) PCA. This was not possible without lowering the regularisation parameter in the rPCA. Another method, the Sparse Sensor Placement for Optimisation (SSPOC), was used to determine the locations in the frequency spectrum that contained the largest differences between structural states. For both the first and second measurement campaigns, these locations were at specific natural frequencies of the system.