generalized eigenspace
When an operator is defective, its ordinary eigenspaces are too small to fill the space. The generalized eigenspace is the repair: instead of stopping at vectors killed by (A - lambda I), you also gather vectors killed by some power (A - lambda I)^k. These extra vectors plug exactly the holes the missing eigenvectors left behind.
Formally, the generalized eigenspace of lambda is G_lambda = ker(A - lambda I)^k for k large enough that the kernel stops growing (k equal to the algebraic multiplicity always suffices). Its dimension equals the algebraic multiplicity of lambda — not merely the geometric multiplicity — so generalized eigenspaces are always 'big enough'. They contain the ordinary eigenspace as the k = 1 layer.
The decisive theorem: over C the whole space is the direct sum of the generalized eigenspaces, V = G_lambda_1 (+) ... (+) G_lambda_m. This always works, even when A is defective and cannot be diagonalized. Choosing a smart basis (Jordan chains) inside each generalized eigenspace yields the Jordan canonical form, the universal normal form for any complex matrix.
Generalized eigenspaces grow until their dimension matches the algebraic multiplicity, filling the defect.
The smallest k for which ker(A - lambda I)^k stabilizes is the size of the largest Jordan block for lambda (the index of the eigenvalue). For a diagonalizable matrix k = 1 everywhere, and generalized eigenspaces collapse back to ordinary eigenspaces.