Vibration-based structural health monitoring (SHM) seeks to detect, quantify, locate, and prognosticate damage by processing vibration signals measured while the structure is operational. The basic premise of vibration-based SHM is that damage will affect the stiffness, mass or energy dissipation properties of the structure and in turn alter its measured dynamic characteristics. In order to make SHM a practical technology it is necessary to perform damage assessment using only a minimum number of permanently installed sensors. Deducing damage at unmeasured regions of the structural domain requires solving an inverse problem that is underdetermined and(or) ill-conditioned. In addition, the effects of local damage on global vibration response may be overshadowed by the effects of modelling error, environmental changes, sensor noise, and unmeasured excitation. These theoretical and practical challenges render the damage identification inverse problem ill-posed, and in some cases unsolvable with conventional inverse methods. This dissertation proposes and tests a novel interpretation of the damage identification inverse problem. Since damage is inherently local and strictly reduces stiffness and(or) mass, the underdetermined inverse problem can be made uniquely solvable by either imposing sparsity or non-negativity on the solution space. The goal of this research is to leverage this concept in order to prove that damage identification can be performed in practical applications using significantly less measurements than conventional inverse methods require. This dissertation investigates two sparsity inducing methods, L1-norm optimization and the non-negative least squares, in their application to identifying damage from eigenvalues, a minimal sensor-based feature that results in an underdetermined inverse problem. This work presents necessary conditions for solution uniqueness and a method to quantify the bounds on the non-unique solution space. The proposed methods are investigated using a wide range of numerical simulations and validated using a four-story lab-scale frame and a full-scale 17 m long aluminum truss. The findings of this study suggest that leveraging the attributes of both L1-norm optimization and non-negative constrained least squares can provide significant improvement over their standalone applications and over other existing methods of damage detection.