# Nilpotent matrix

A **nilpotent matrix** is a square matrix for which some finite power is the zero matrix: there exists a positive integer k with \( A^{k} = 0 \). The least such k is called the index of nilpotency (or nilpotency index)<sup>[1](https://www.uvm.edu/~cvincen1/files/teaching/fall2016-math124/book.pdf)</sup><sup> • </sup><sup>[2](https://encyclopediaofmath.org/wiki/Nilpotent_element)</sup>. Nilpotent matrices are the building blocks from which the Jordan canonical form of an arbitrary square matrix is assembled, and they mark exactly the part of a linear operator that eigenvalues and eigenvectors fail to describe<sup>[3](https://linear.pugetsound.edu/scla/html/section-jordan-canonical-form.html)</sup><sup> • </sup><sup>[4](https://www.math.uwaterloo.ca/~cgodsil/quagmire/Fields24/pdfs/LA2.pdf)</sup>.

| Key fact | Detail |
|---|---|
| Definition | \( A^{k} = 0 \) for some positive integer k; the least such k is the index of nilpotency<sup>[1](https://www.uvm.edu/~cvincen1/files/teaching/fall2016-math124/book.pdf)</sup> |
| Index bound | For an \( n \times n \) nilpotent matrix the index is at most n<sup>[5](https://www-personal.umd.umich.edu/~adwiggin/TeachingFiles/LinearAlgebra227/Exams/Math227FinalWinter20.pdf)</sup> |
| Eigenvalues | 0 is the only eigenvalue; the minimal polynomial is \( t^{k} \), where k is the index<sup>[4](https://www.math.uwaterloo.ca/~cgodsil/quagmire/Fields24/pdfs/LA2.pdf)</sup> |
| Canonical form | Every nilpotent matrix is similar to a block diagonal matrix of nilpotent Jordan blocks \( N_{k} \)<sup>[4](https://www.math.uwaterloo.ca/~cgodsil/quagmire/Fields24/pdfs/LA2.pdf)</sup> |
| Exponential | \( e^{A} = \sum_{i=0}^{k-1} A^{i}/i! \) is a finite sum, and exp is a bijective polynomial map from nilpotent matrices to unipotent matrices<sup>[6](https://webhomes.maths.ed.ac.uk/~v1ranick/papers/goodwallx.pdf)</sup> |
| Graph meaning | A directed graph is acyclic if and only if its weighted adjacency matrix is nilpotent<sup>[7](https://papers.neurips.cc/paper_files/paper/2020/file/28a7602724ba16600d5ccc644c19bf18-Paper.pdf)</sup> |

## Definition and basic examples

The defining condition is that repeated multiplication eventually annihilates the matrix. A nilpotent matrix of index 1 is the zero matrix itself<sup>[4](https://www.math.uwaterloo.ca/~cgodsil/quagmire/Fields24/pdfs/LA2.pdf)</sup>. The UC Davis linear algebra glossary states the same idea informally: a nilpotent matrix is one for which the square, the cube, or some finite power equals zero<sup>[8](https://www.math.ucdavis.edu/~daddel/MATH22AL/Resources/Linear_Algebra_Glossary.html)</sup>.

Familiar examples come from shifts. The companion matrix of the polynomial \( t^{k} \) is a nilpotent matrix of index k, with ones on the subdiagonal or superdiagonal and zeros elsewhere; in one common basis it satisfies \( N_{k} e_{1} = 0 \) and \( N_{k} e_{i+1} = e_{i} \)<sup>[4](https://www.math.uwaterloo.ca/~cgodsil/quagmire/Fields24/pdfs/LA2.pdf)</sup>. Nicolaescu's linear algebra notes call this matrix a (nilpotent) Jordan cell of size n<sup>[9](https://www3.nd.edu/~lnicolae/Linalg_notes.pdf)</sup>. A two-by-two example is the matrix with a 1 below the diagonal and zeros elsewhere, whose square is zero<sup>[4](https://www.math.uwaterloo.ca/~cgodsil/quagmire/Fields24/pdfs/LA2.pdf)</sup>.

Nilpotency also appears outside matrices proper. The Encyclopedia of Mathematics notes that in the ring of \( 2 \times 2 \) matrices over a field K a strictly triangular matrix of a standard form is nilpotent of index 2, and that in the group algebra \( F_{p}[G] \) of the cyclic group of order p the element \( 1 - \sigma \) is nilpotent of index p<sup>[2](https://encyclopediaofmath.org/wiki/Nilpotent_element)</sup>. On function spaces, the differentiation map \( d/dx \) on the polynomials of degree at most 2 is nilpotent of index three, since the third derivative of any quadratic polynomial is zero<sup>[1](https://www.uvm.edu/~cvincen1/files/teaching/fall2016-math124/book.pdf)</sup>.

## Eigenvalues and what they miss

A nilpotent matrix has 0 as its only eigenvalue. If \( \lambda \neq 0 \) were an eigenvalue with eigenvector x, then \( A^{k}x = \lambda^{k} x \neq 0 \), contradicting \( A^{k} = 0 \)<sup>[4](https://www.math.uwaterloo.ca/~cgodsil/quagmire/Fields24/pdfs/LA2.pdf)</sup>. The minimal polynomial of a nilpotent matrix of index k is \( t^{k} \)<sup>[4](https://www.math.uwaterloo.ca/~cgodsil/quagmire/Fields24/pdfs/LA2.pdf)</sup>.

**Eigenvalues underdetermine the matrix.** For the companion matrix \( N_{k} \), the rank is \( k - 1 \), so the eigenspace for 0 has dimension 1 and consists of the span of \( e_{1} \); Godsil's notes conclude that eigenvalues and eigenvectors provide very little information about nilpotent matrices<sup>[4](https://www.math.uwaterloo.ca/~cgodsil/quagmire/Fields24/pdfs/LA2.pdf)</sup>. Two nilpotent matrices of the same size can share the entire spectrum, the zero matrix, and differ completely in behavior.

## Canonical form

The structure theorem states that every nilpotent matrix is similar to a block diagonal matrix in which each diagonal block is a nilpotent companion matrix \( N_{k} \) for some k; Godsil identifies this block diagonal form as the Frobenius normal form<sup>[4](https://www.math.uwaterloo.ca/~cgodsil/quagmire/Fields24/pdfs/LA2.pdf)</sup>. In the Jordan formulation, a nilpotent operator admits a Jordan basis of disjoint towers of vectors, and the restriction to each tower is represented by a nilpotent Jordan cell; the heights of the towers are invariants of the operator, the same in every Jordan basis<sup>[9](https://www3.nd.edu/~lnicolae/Linalg_notes.pdf)</sup>.

The block sizes classify nilpotent matrices up to similarity: the number of similarity classes of \( n \times n \) nilpotent matrices over a field equals the number of integer partitions of n, recorded as nonincreasing tuples \( (k_{1}, \ldots, k_{r}) \) with \( k_{1} + \cdots + k_{r} = n \)<sup>[4](https://www.math.uwaterloo.ca/~cgodsil/quagmire/Fields24/pdfs/LA2.pdf)</sup>. The largest block size equals the index of nilpotency; the SCLA text illustrates this with an operator of index 4 and nullity 2, whose form consists of exactly two Jordan blocks, one of size 4 and one of size 2<sup>[3](https://linear.pugetsound.edu/scla/html/section-jordan-canonical-form.html)</sup>.

The index cannot exceed the size of the matrix. For an \( n \times n \) nilpotent matrix the largest possible [Jordan block](https://www.edgepedia.org/jordan-block) is \( n \times n \), and multiplying such a block by itself n times yields zero, so the nilpotency index is never greater than n<sup>[5](https://www-personal.umd.umich.edu/~adwiggin/TeachingFiles/LinearAlgebra227/Exams/Math227FinalWinter20.pdf)</sup>.

Nilpotent blocks also sit inside general matrices. By the primary decomposition theorem, a matrix over an algebraically closed field with minimal polynomial \( \psi(t) \) is similar to a block diagonal matrix with blocks \( \theta I + N_{\theta} \), where \( \theta \) runs over the zeros of \( \psi \) and each \( N_{\theta} \) is nilpotent with index equal to the multiplicity of \( \theta \) as a zero of \( \psi \)<sup>[4](https://www.math.uwaterloo.ca/~cgodsil/quagmire/Fields24/pdfs/LA2.pdf)</sup>. This is the Jordan canonical form, proposed by the French mathematician [Camille Jordan](https://www.edgepedia.org/camille-jordan) in his 1870 treatise *Traité des substitutions et des équations algébriques*, initially over a finite field rather than the complex numbers<sup>[10](https://mathshistory.st-andrews.ac.uk/Biographies//Jordan/)</sup><sup> • </sup><sup>[11](https://doi.org/10.1016%2Fj.laa.2021.12.014)</sup>.

## Powers, exponentials, and transient growth

Powers of a nilpotent matrix terminate, which makes several formulas finite. The exponential series truncates: for a nilpotent matrix of index k,

\[ e^{A} = \sum_{i=0}^{k-1} \frac{A^{i}}{i!}. \]

The same truncation underlies the formal exponential map used in nilpotent Lie algebras, where the series is cut off because all terms with exponents larger than the nilpotency bound vanish<sup>[12](https://msl.cs.illinois.edu/~lavalle/planning/node840.html)</sup>. More strongly, the exponential function is a bijective polynomial map from the nilpotent elements of \( M_{n}(\mathbb{C}) \) onto the unipotent elements of \( GL(n, \mathbb{C}) \), with inverse given by the logarithm \( u \mapsto \log u \)<sup>[6](https://webhomes.maths.ed.ac.uk/~v1ranick/papers/goodwallx.pdf)</sup>. On simply connected nilpotent Lie groups this finiteness has global consequences: the exponential map is a diffeomorphism and the group product in exponential coordinates is polynomial<sup>[13](https://link.springer.com/chapter/10.1007/978-3-031-98832-5_9)</sup>.

**Nilpotent parts can dominate finite-time behavior.** A numerical linear algebra essay examines the \( 6 \times 6 \) matrix \( A = 0.8 \cdot I + m \cdot N \), where N is the nilpotent shift with \( N^{6} = 0 \). The spectrum is \( \{0.8\} \) with multiplicity six at every value of m, yet at \( m = 2 \) the norm of \( A^{k} \) rises by four orders of magnitude before decaying. Because the identity commutes with N, the binomial theorem applies and terminates:

\[ A^{k} = \sum_{j=0}^{5} \binom{k}{j} \cdot 0.8^{\,k-j} \cdot m^{j} \cdot N^{j}. \]

The largest term is \( \binom{k}{5} \cdot 0.8^{\,k-5} \cdot m^{5} \), a degree-5 polynomial in k multiplied by a geometric decay, which rises and then falls; the turning point is about \( k = (n-1)/(1-\rho) \), which for \( n = 6 \) and \( \rho = 0.8 \) is 25, with the measured peak at step 24. The essay attributes the transient to the \( n - 1 \) binomial coefficients, the cost of a defective eigenvalue showing up in the powers rather than in the eigenvectors<sup>[14](https://www.numerical-linear-algebra.com/essays/a-spectral-radius-that-grows-first/)</sup>.

## Applications

**Directed acyclic graphs.** The powers of an adjacency matrix count directed walks, so nilpotency of the adjacency matrix is equivalent to the absence of cycles. A NeurIPS paper on learning Bayesian networks uses this as Lemma 1: a directed graph is acyclic if and only if its weighted adjacency matrix is nilpotent, equivalently if \( \mathrm{tr}(A^{p}) = 0 \) for all p, and builds a differentiable acyclicity constraint on this characterization<sup>[7](https://papers.neurips.cc/paper_files/paper/2020/file/28a7602724ba16600d5ccc644c19bf18-Paper.pdf)</sup>. Related work reduces the Maximal Acyclic Subgraph problem to finding the closest nilpotent matrix to a graph's matrix<sup>[15](https://arxiv.org/abs/1912.03783)</sup>.

**Control theory.** Nilpotent system matrices support fixed-time stabilization results: for a precisely known linear system with a nilpotent matrix, a feedback using an additional artificial delay drives the state to zero in fixed time and achieves input-to-state stability under additive disturbances and small perturbations<sup>[16](https://arxiv.org/abs/2607.24595)</sup>. Time-optimal control of systems whose matrix A is nilpotent with nilpotency index n has been treated as a binary classification problem for the bang-bang feedback law<sup>[17](https://link.springer.com/article/10.1007/s42967-026-00619-1)</sup>. Nilpotent approximations of vector fields, in which the approximating system generates a nilpotent [Lie algebra](https://www.edgepedia.org/lie-algebra), are a constructive tool in nonlinear control<sup>[18](https://epubs.siam.org/doi/abs/10.1137/0324045)</sup>.

**Operator theory.** The numerical range of a nilpotent matrix is not fully characterized in general, but a \( 3 \times 3 \) nilpotent matrix M has a circular numerical range if and only if \( \mathrm{tr}(M^{*}M^{2}) = 0 \), in which case the numerical radius is \( w(M) = \sqrt{\mathrm{tr}(M^{*}M)}/2 \)<sup>[19](https://www.sciencedirect.com/science/article/abs/pii/S0096300310000652)</sup>.

## Related notions

Nilpotency extends beyond matrices. A Lie algebra is nilpotent when its descending central series terminates, and a [Lie group](https://www.edgepedia.org/lie-group) is nilpotent when its Lie algebra is<sup>[20](https://onlinelibrary.wiley.com/doi/10.1155/2019/2963120)</sup>. In ring theory, an element a is nilpotent when some power \( a^{n} = 0 \), with the nilpotency index defined the same way as for matrices; the residue class of p in the ring modulo \( p^{n} \) is nilpotent of index n<sup>[2](https://encyclopediaofmath.org/wiki/Nilpotent_element)</sup>. Within matrix theory, nilpotent matrices are the opposite extreme from diagonalizable ones: a matrix is diagonalizable if and only if its minimal polynomial consists of distinct linear factors, while a nilpotent matrix of index k has the repeated-root minimal polynomial \( t^{k} \)<sup>[8](https://www.math.ucdavis.edu/~daddel/MATH22AL/Resources/Linear_Algebra_Glossary.html)</sup><sup> • </sup><sup>[4](https://www.math.uwaterloo.ca/~cgodsil/quagmire/Fields24/pdfs/LA2.pdf)</sup>.

## References

1. [Linear Algebra (Vincenzi, University of Vermont), definition and index of nilpotency](https://www.uvm.edu/~cvincen1/files/teaching/fall2016-math124/book.pdf)
2. [Nilpotent element, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Nilpotent_element)
3. [Section 3.3 Jordan Canonical Form, SCLA (University of Puget Sound)](https://linear.pugetsound.edu/scla/html/section-jordan-canonical-form.html)
4. [More Linear Algebra (Godsil, University of Waterloo)](https://www.math.uwaterloo.ca/~cgodsil/quagmire/Fields24/pdfs/LA2.pdf)
5. [Math 227 Final Exam, Winter 2020, University of Michigan-Dearborn (index at most n)](https://www-personal.umd.umich.edu/~adwiggin/TeachingFiles/LinearAlgebra227/Exams/Math227FinalWinter20.pdf)
6. [Lie Groups and Algebraic Groups (Ranicki notes), exponential map on nilpotent elements](https://webhomes.maths.ed.ac.uk/~v1ranick/papers/goodwallx.pdf)
7. [DAG-GNN / acyclicity characterization, NeurIPS 2020 paper](https://papers.neurips.cc/paper_files/paper/2020/file/28a7602724ba16600d5ccc644c19bf18-Paper.pdf)
8. [Linear Algebra Glossary, UC Davis MATH22AL](https://www.math.ucdavis.edu/~daddel/MATH22AL/Resources/Linear_Algebra_Glossary.html)
9. [Notes on Linear Algebra (Liviu I. Nicolaescu, University of Notre Dame)](https://www3.nd.edu/~lnicolae/Linalg_notes.pdf)
10. [Camille Jordan (1838–1922), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies//Jordan/)
11. [The numerical Jordan form, Linear Algebra and its Applications (2022)](https://doi.org/10.1016%2Fj.laa.2021.12.014)
12. [The exponential map, Planning Algorithms (LaValle), University of Illinois](https://msl.cs.illinois.edu/~lavalle/planning/node840.html)
13. [Nilpotent Lie Groups, Springer Nature chapter](https://link.springer.com/chapter/10.1007/978-3-031-98832-5_9)
14. [A spectral radius that grows first, Numerical Linear Algebra essays](https://www.numerical-linear-algebra.com/essays/a-spectral-radius-that-grows-first/)
15. [Maximal acyclic subgraphs and closest stable matrices, arXiv:1912.03783](https://arxiv.org/abs/1912.03783)
16. [Fixed-time stabilization of systems with nilpotent matrices in the presence of large unknown delays, arXiv:2607.24595](https://arxiv.org/abs/2607.24595)
17. [Time-Optimal Neural Feedback Control of Nilpotent Systems as a Binary Classification Problem, Communications on Applied Mathematics and Computation](https://link.springer.com/article/10.1007/s42967-026-00619-1)
18. [Nilpotent Approximations of Control Systems and Distributions, SIAM Journal on Control and Optimization](https://epubs.siam.org/doi/abs/10.1137/0324045)
19. [When is the numerical range of a nilpotent matrix circular?, Linear Algebra and its Applications](https://www.sciencedirect.com/science/article/abs/pii/S0096300310000652)
20. [Control Sets of Linear Control Systems on Matrix Groups and Applications, Mathematical Problems in Engineering (2019)](https://onlinelibrary.wiley.com/doi/10.1155/2019/2963120)

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