Generalized eigenvector

A generalized eigenvector of a linear operator $A$ for a scalar $\lambda$ is a nonzero vector $v$ with $(A-\lambda I{)}^{k}v=0$ for some positive integer $k$. An ordinary eigenvector is the case $k=1$.

For $A=\left(\begin{array}{cc}\lambda & 1\\ 0& \lambda \end{array}\right)$, the second coordinate vector $e_2$ is generalized: $(A-\lambda I)e_2=e_1$ and $(A-\lambda I{)}^2{e}_2=0$, although it is not an eigenvector. Such vectors form Jordan chains and complete an eigenvector set when the operator is not diagonalizable. Excluding the zero vector is part of the definition.