Assess students' understanding of diagonalization for real n×n matrices: computing eigenvalues and eigenvectors, distinguishing algebraic and geometric multiplicities, criteria for diagonalizability over R, constructing similarity transformations (P−1AP = D) and change-of-basis matrices, recognizing symmetric and defective cases, and applying diagonalization to compute matrix powers; tasks include determining diagonalizability, producing P and D when possible, and explaining failures.