defective operator
Most of the time a matrix has a full set of eigenvectors — enough independent ones to build a basis and diagonalize. A defective operator is the troublemaker that falls short: it does not have enough eigenvectors to form a basis, so it simply cannot be diagonalized, no matter how cleverly you choose coordinates.
The precise diagnosis is a multiplicity gap. An operator is defective exactly when some eigenvalue has geometric multiplicity strictly less than its algebraic multiplicity: dim ker(A - lambda I) < (the order of lambda as a root). The missing eigenvectors are the defect. The classic small example is [2, 1; 0, 2], where lambda = 2 has algebraic multiplicity 2 but only a one-dimensional eigenspace.
Defectiveness is not a dead end — it is the doorway to the Jordan form. To compensate for the missing eigenvectors, you bring in generalized eigenvectors from ker(A - lambda I)^k, which together do span the space. The matrix then becomes block-diagonal with Jordan blocks: nearly diagonal, with a few 1s sitting just above the diagonal to record the defect.
Both eigenvalues are 2, but every eigenvector is a multiple of (1, 0): one direction short, hence defective.
Defectiveness is fragile: an arbitrarily tiny perturbation of a defective matrix usually makes the eigenvalues distinct and the matrix diagonalizable. Defective matrices form a measure-zero set, yet they appear constantly in honest models (repeated roots, resonance).