# Generalized eigenvector
A **generalized eigenvector** of a square matrix \( A \) with eigenvalue \( \lambda \) is a nonzero vector \( \mathbf{v} \) such that \( (A - \lambda I)^{p} \mathbf{v} = \mathbf{0} \) for some positive integer \( p \).[1](https://mathbooks.unl.edu/DifferentialEquations/linear09.html) Ordinary eigenvectors are the special case \( p = 1 \).
| Key fact | Detail |
|---|---|
| Definition | Nonzero \( \mathbf{v} \) with \( (A - \lambda I)^{p} \mathbf{v} = \mathbf{0} \) for some \( p \geq 1 \); ordinary eigenvectors correspond to \( p = 1 \)[1](https://mathbooks.unl.edu/DifferentialEquations/linear09.html) |
| Generalized eigenspace | \( E^{g}_{\lambda}(A) = N\left((A - \lambda I)^{m_{a}(\lambda)}\right) \), where \( m_{a}(\lambda) \) is the algebraic multiplicity; its nonzero elements are the generalized eigenvectors[3](https://ximera.osu.edu/linearalgebra/textbook/eigenvalueProperties/generalizedEigenvectors) |
| Filtration | \( E^{1}_{\lambda}(A) \subset E^{2}_{\lambda}(A) \subset \cdots \subset E^{m_{a}(\lambda)}_{\lambda}(A) = E^{g}_{\lambda}(A) \); once a kernel stops growing it stays fixed for all higher powers[3](https://ximera.osu.edu/linearalgebra/textbook/eigenvalueProperties/generalizedEigenvectors) • [4](https://runestone.uleth.ca/ns/books/published/linearpython/sec-gen-eigen.html) |
| Decomposition | For distinct eigenvalues \( \lambda_1, \ldots, \lambda_k \) of an \( n \times n \) matrix, the generalized eigenspaces form a direct sum equal to \( \mathbb{C}^{n} \)[3](https://ximera.osu.edu/linearalgebra/textbook/eigenvalueProperties/generalizedEigenvectors) |
| Multiplicity relation | The algebraic multiplicity of \( \lambda \) equals the maximum number of linearly independent generalized \( \lambda \)-eigenvectors[5](https://www.math.clemson.edu/~macaule/classes/f20_math8530/slides/math8530_lecture-4-07_h.pdf) |
| Jordan blocks | The number of Jordan blocks for \( \lambda \) is its geometric multiplicity; the sum of their sizes is its algebraic multiplicity[6](https://www.math.ucla.edu/~njhu/notes/quals-ubc/diffeqs/) |
| Main use | Completing a basis of generalized eigenvectors puts \( A \) in Jordan normal form, and supports solving \( \mathbf{x}' = A\mathbf{x} \) via the matrix exponential[2](https://alayont-f.github.io/linalg/chap_JCF.html) • [7](https://www.youtube.com/watch?v=HOK2p_vy3rI) |
## Definition and rank
The definition extends the eigenvector equation by allowing powers of \( A - \lambda I \). Given a vector \( \mathbf{v} \) satisfying \( (A - \lambda I)^{p} \mathbf{v} = \mathbf{0} \) for some integer \( p \geq 1 \), where \( \lambda \) is an eigenvalue of \( A \), the vector \( \mathbf{v} \) is called a generalized eigenvector of \( A \).[1](https://mathbooks.unl.edu/DifferentialEquations/linear09.html) Equivalently, a nonzero element of the generalized eigenspace \( E^{g}_{\lambda}(A) \) is called a generalized eigenvector.[3](https://ximera.osu.edu/linearalgebra/textbook/eigenvalueProperties/generalizedEigenvectors)
Every generalized eigenvector has a well-defined **rank**: the unique integer \( m \geq 1 \) for which \( (A - \lambda I)^{m} \mathbf{v} = \mathbf{0} \) while \( (A - \lambda I)^{m-1} \mathbf{v} \neq \mathbf{0} \).[3](https://ximera.osu.edu/linearalgebra/textbook/eigenvalueProperties/generalizedEigenvectors) [Rank 1](https://www.edgepedia.org/rank-1) generalized eigenvectors are exactly the ordinary eigenvectors. Applying \( A - \lambda I \) to a rank-\( m \) vector lowers its rank by one, which is the mechanism that organizes generalized eigenvectors into chains.
## The generalized eigenspace
For each \( k \), define \( E^{k}_{\lambda}(A) = N\left((A - \lambda I)^{k}\right) \), the kernel of the \( k \)-th power. These kernels form an increasing sequence
\[ E_{\lambda}(A) = E^{1}_{\lambda}(A) \subset E^{2}_{\lambda}(A) \subset \cdots \subset E^{m_{a}(\lambda)}_{\lambda}(A) = E^{g}_{\lambda}(A). \]
The kernels can grow for a while, but the moment the kernel does not change from one power to the next, it stops changing for all further powers.[4](https://runestone.uleth.ca/ns/books/published/linearpython/sec-gen-eigen.html) Taking the power equal to the algebraic multiplicity \( m_{a}(\lambda) \) is always enough, so the generalized eigenspace can be computed as a single null space rather than by testing every power.[3](https://ximera.osu.edu/linearalgebra/textbook/eigenvalueProperties/generalizedEigenvectors) • [8](https://linear.pugetsound.edu/scla/html/section-generalized-eigenspaces.html) In practice this means putting \( (A - \lambda I)^{m_a} \) in echelon form and reading off its nullspace, just as for an ordinary eigenspace.[9](https://strncat.github.io/jekyll/update/2024/09/15/lec37-jordan-canonical-form.html)
The generalized eigenspaces of distinct eigenvalues are independent of one another. If \( \lambda_1, \lambda_2, \ldots, \lambda_k \) are the distinct eigenvalues of an \( n \times n \) matrix \( A \), then
\[ E^{g}_{\lambda_1}(A) \oplus E^{g}_{\lambda_2}(A) \oplus \cdots \oplus E^{g}_{\lambda_k}(A) = \mathbb{C}^{n}. \]
Joining bases of all the generalized eigenspaces therefore produces a basis of the whole space.[3](https://ximera.osu.edu/linearalgebra/textbook/eigenvalueProperties/generalizedEigenvectors) • [10](https://www.math.brown.edu/streil/papers/LADW/HTML_2026_08-31/Ch9.html) This is the sense in which generalized eigenvectors repair the failure of ordinary eigenvectors to span: even a defective matrix has enough of them to form a full basis.
## Jordan chains
Generalized eigenvectors of a given eigenvalue are organized into **Jordan chains**. If \( \mathbf{v} \) is a rank-\( k \) generalized eigenvector of \( T \) for the eigenvalue \( \lambda \), the chain generated by \( \mathbf{v} \) is
\[ \mathbf{v}_k, \mathbf{v}_{k-1}, \ldots, \mathbf{v}_1, \qquad \mathbf{v}_j = (T - \lambda I)^{k-j} \mathbf{v}, \]
where each \( \mathbf{v}_j \) is a rank-\( j \) generalized eigenvector.[6](https://www.math.ucla.edu/~njhu/notes/quals-ubc/diffeqs/) The vectors in a chain are linearly independent, and the last one, \( (A - \lambda I)^{l-1} \mathbf{v}_m \), is an ordinary eigenvector.[11](https://sites.math.rutgers.edu/~zchan/current-preprint/642-550Notes.pdf) By definition of rank, every vector in a Jordan chain is nonzero.[3](https://ximera.osu.edu/linearalgebra/textbook/eigenvalueProperties/generalizedEigenvectors)
A concrete example shows how a chain fills a missing basis slot. For a \( 3 \times 3 \) matrix with eigenvalue \( \lambda = 5 \) of algebraic multiplicity 2 and geometric multiplicity 1, there is only one ordinary eigenvector for \( \lambda = 5 \), but a second vector that is killed by \( (T - 5I)^{2} \) though not by \( T - 5I \) completes the basis. In that basis the matrix takes the form with a \( 2 \times 2 \) [Jordan block](https://www.edgepedia.org/jordan-block), having the eigenvalue 5 on the diagonal and a 1 directly above it.[2](https://alayont-f.github.io/linalg/chap_JCF.html) • [12](https://jovanaeducation.com/glossary/generalized-eigenvector)
## Jordan normal form
The purpose of collecting Jordan chains is the **Jordan normal form** of the matrix. When a basis is built from Jordan chains, the matrix of \( T \) in that basis consists of Jordan blocks, each block carrying a single eigenvalue on its diagonal and 1s on the superdiagonal.[6](https://www.math.ucla.edu/~njhu/notes/quals-ubc/diffeqs/) The block structure records the multiplicities: the number of Jordan blocks with \( \lambda_i \) on the diagonal is the geometric multiplicity of \( \lambda_i \), and the sum of the sizes of those blocks is its algebraic multiplicity.[6](https://www.math.ucla.edu/~njhu/notes/quals-ubc/diffeqs/)
The algebraic multiplicity also has a direct interpretation in terms of generalized eigenvectors: it equals the maximum number of linearly independent generalized \( \lambda \)-eigenvectors of \( A \).[5](https://www.math.clemson.edu/~macaule/classes/f20_math8530/slides/math8530_lecture-4-07_h.pdf) On each generalized eigenspace the restricted operator \( N_k = A - \lambda_k I \) is nilpotent, with \( N_k^{d_k} = \mathbf{0} \), which is what makes the block structure possible.[10](https://www.math.brown.edu/streil/papers/LADW/HTML_2026_08-31/Ch9.html)
## Applications
The most common application is solving systems of linear differential equations. For \( \mathbf{x}'(t) = A\mathbf{x}(t) \), the solution involves the matrix exponential \( e^{tA} \). The same structure underlies closed-form formulas for matrix powers, the matrix exponential, and spectral projections.[13](https://arxiv.org/abs/2010.12551)
The eigenscheme viewpoint records this information geometrically: the eigenscheme of a matrix encodes information about both its eigenvectors and its generalized eigenvectors, and its decomposition encodes the numeric data of the Jordan canonical form.[14](https://arxiv.org/abs/1506.08257)
## References
1. [The Matrix Exponential, ODE Project (University of Nebraska Lincoln)](https://mathbooks.unl.edu/DifferentialEquations/linear09.html)
2. [Jordan Normal Form: When a Matrix Can't Be Diagonalized, DataField](https://alayont-f.github.io/linalg/chap_JCF.html)
3. [Generalized Eigenvectors, Ximera (Ohio State University)](https://ximera.osu.edu/linearalgebra/textbook/eigenvalueProperties/generalizedEigenvectors)
4. [Generalized eigenspaces, Runestone (University of Lethbridge)](https://runestone.uleth.ca/ns/books/published/linearpython/sec-gen-eigen.html)
5. [Lecture 4.7: Jordan canonical form, M. Macauley, Clemson University](https://www.math.clemson.edu/~macaule/classes/f20_math8530/slides/math8530_lecture-4-07_h.pdf)
6. [Differential Equations notes, Nicholas Hu, UCLA](https://www.math.ucla.edu/~njhu/notes/quals-ubc/diffeqs/)
7. [Jordan Canonical Form 2x2 Case, Generalized Eigenvectors (video)](https://www.youtube.com/watch?v=HOK2p_vy3rI)
8. [SCLA Generalized Eigenspaces, Puget Sound](https://linear.pugetsound.edu/scla/html/section-generalized-eigenspaces.html)
9. [Lecture 37/38: The Jordan Canonical Form and Generalized Eigenvectors](https://strncat.github.io/jekyll/update/2024/09/15/lec37-jordan-canonical-form.html)
10. [Chapter 9: Advanced spectral theory, Linear Algebra Done Wrong, S. Treil, Brown University](https://www.math.brown.edu/streil/papers/LADW/HTML_2026_08-31/Ch9.html)
11. [Notes and Comments for Linear Algebra and Its Applications, Rutgers](https://sites.math.rutgers.edu/~zchan/current-preprint/642-550Notes.pdf)
12. [Generalized eigenvector, JOVANA Glossary](https://jovanaeducation.com/glossary/generalized-eigenvector)
13. [Some Explicit Formulas for Matrix Exponential, Matrix Logarithm, the nth Power of Matrices and their Drazin Inverses, arXiv](https://arxiv.org/abs/2010.12551)
14. [Eigenschemes and the Jordan canonical form, arXiv](https://arxiv.org/abs/1506.08257)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Linear and multilinear algebra › Decompositions and canonical forms › Jordan and rational canonical forms*
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