A defective matrix is a square matrix that does not have a complete set of linearly independent eigenvectors. This means that its algebraic multiplicity (the number of times an eigenvalue occurs as a root of the characteristic polynomial) is greater than its geometric multiplicity (the number of linearly independent eigenvectors associated with that eigenvalue). In other words, a matrix is considered defective if it cannot be diagonalized.
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A matrix is defective when an eigenvalue has geometric multiplicity smaller than its algebraic multiplicity. Over a field in which the characteristic polynomial splits, this is precisely the obstruction to an eigenbasis: at least one Jordan block has size greater than one, and generalized eigenvectors are needed to complete a basis. The shear mapping in diagonalizable matrix is the simplest example.