Assess undergraduate students' understanding of the geometric meaning of eigenvalues and eigenvectors and the algebraic criteria for matrix diagonalization. Tasks should include interpreting eigenpairs as invariant directions and directional scalings, distinguishing algebraic vs geometric multiplicity, deciding diagonalizability by testing for an eigenbasis (including implications of repeated eigenvalues), applying the spectral theorem for symmetric matrices (orthogonal diagonalization), and computing eigenvectors/eigenspaces for specific matrices to determine diagonalizability over R or C; include short proofs, geometric explanations, and constructive diagonalizations or counterexamples as appropriate.