Generalized eigenvector

In linear algebra, a generalized eigenvector of an ${\displaystyle n\times n}$ matrix ${\displaystyle A}$ is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector.^[1]

Let ${\displaystyle V}$ be an ${\displaystyle n}$-dimensional vector space and let ${\displaystyle A}$ be the matrix representation of a linear map from ${\displaystyle V}$ to ${\displaystyle V}$ with respect to some ordered basis.

There may not always exist a full set of ${\displaystyle n}$ linearly independent eigenvectors of ${\displaystyle A}$ that form a complete basis for ${\displaystyle V}$. That is, the matrix ${\displaystyle A}$ may not be diagonalizable.^[2][3] This happens when the algebraic multiplicity of at least one eigenvalue ${\displaystyle \lambda _{i}}$ is greater than its geometric multiplicity (the nullity of the matrix ${\displaystyle (A-\lambda _{i}I)}$, or the dimension of its nullspace). In this case, ${\displaystyle \lambda _{i}}$ is called a defective eigenvalue and ${\displaystyle A}$ is called a defective matrix.^[4]

A generalized eigenvector ${\displaystyle x_{i}}$ corresponding to ${\displaystyle \lambda _{i}}$, together with the matrix ${\displaystyle (A-\lambda _{i}I)}$ generate a Jordan chain of linearly independent generalized eigenvectors which form a basis for an invariant subspace of ${\displaystyle V}$.^[5][6][7]

Using generalized eigenvectors, a set of linearly independent eigenvectors of ${\displaystyle A}$ can be extended, if necessary, to a complete basis for ${\displaystyle V}$.^[8] This basis can be used to determine an "almost diagonal matrix" ${\displaystyle J}$ in Jordan normal form, similar to ${\displaystyle A}$, which is useful in computing certain matrix functions of ${\displaystyle A}$.^[9] The matrix ${\displaystyle J}$ is also useful in solving the system of linear differential equations ${\displaystyle \mathbf {x} '=A\mathbf {x} ,}$ where ${\displaystyle A}$ need not be diagonalizable.^[10][11]

The dimension of the generalized eigenspace corresponding to a given eigenvalue ${\displaystyle \lambda }$ is the algebraic multiplicity of ${\displaystyle \lambda }$.^[12]