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This paper introduces a methodology for improving the accuracy and efficiency of reduced order models (ROMs) constructed using the least-squares Petrov-Galerkin (LSPG) projection method through the introduction of preconditioning. Unlike prior related work, which focuses on preconditioning the linear systems arising within the ROM numerical solution procedure to improve linear solver performance, our approach leverages a preconditioning matrix directly within the LSPG minimization problem. Applying preconditioning in this way can improve ROM accuracy for several reasons. First, preconditioning the LSPG formulation changes the norm defining the residual minimization, which can improve the residual-based stability constant bounding the ROM solution's error. The incorporation of a preconditioner into the LSPG formulation can have the additional effect of scaling the components of the residual being minimized, which can be beneficial for problems with disparate scales. Importantly, we demonstrate that an 'ideal preconditioned' LSPG ROM (a ROM preconditioned with the inverse of the Jacobian of its corresponding full order model, or FOM) emulates projection of the FOM solution increment onto the reduced basis, a lower bound on the ROM solution error for a given reduced basis. By designing preconditioners that approximate the Jacobian inverse, a ROM whose error approaches this lower bound can be obtained. The proposed approach is evaluated in the predictive regime on several mechanical and thermo-mechanical problems within the Albany HPC code. We demonstrate numerically that the introduction of simple Jacobi, Gauss-Seidel and ILU preconditioners into the Proper Orthogonal Decomposition/LSPG formulation reduces significantly the ROM solution error, the reduced Jacobian condition number, the number of nonlinear iterations required to reach convergence, and the wall time.