Assess learners' ability to analyze spectral properties of ill-conditioned matrices and design effective preconditioners for Krylov subspace iterative solvers. Scope: diagnosing ill-conditioning via eigenvalues, singular values and pseudospectra; understanding effects on convergence of CG, GMRES, and BiCG; selecting and formulating preconditioners (ILU/IC, multigrid, approximate inverses, polynomial, deflation, block strategies), scaling/equilibration, and using metrics (condition number, eigenvalue clustering, Ritz values) to evaluate performance, cost, and robustness in practical implementations.