Algebraic and Geometric Multiplicities of Eigenvalues

Q: Why can the geometric multiplicity never exceed the algebraic multiplicity?

Because a $g$-dimensional eigenspace can be extended to a basis of $\mathbb{C}^n$, and the resulting similarity transformation puts $g$ columns of the transformed matrix into the form $\lambda \vec{e}_i$. Expanding the determinant down each of those columns extracts $g$ factors of $(x - \lambda)$ from the characteristic polynomial, so the algebraic multiplicity is at least $g$. Similar matrices share a characteristic polynomial, which is what transfers the conclusion back to $A$.

Q: If a matrix is defective, what completes the basis?

Generalised eigenvectors. For an eigenvalue with defect $d$, one solves $(A - \lambda I_n)\vec{w} = \vec{v}$ starting from an existing eigenvector $\vec{v}$ and continues the chain until $\alpha_A(\lambda)$ vectors are associated with $\lambda$. In the resulting basis the matrix takes Jordan form: diagonal blocks with $\lambda$ on the diagonal and ones on the superdiagonal.

Overview

A repeated eigenvalue is the first place where the theory stops being uniform. When every root of the characteristic polynomial is simple, everything follows automatically: one eigenvector per eigenvalue, $n$ of them, independent by the distinct-eigenvalue theorem, and a diagonal representation. As soon as a root repeats, a question opens that the polynomial cannot answer — does the repeated eigenvalue supply as many independent eigenvectors as its repetition count suggests?

The two multiplicities formalise the question. The algebraic multiplicity $\alpha_A(\lambda)$ is read from the factored characteristic polynomial and is a statement about a scalar equation. The geometric multiplicity $\gamma_A(\lambda)$ is the dimension of the eigenspace $\mathcal{N}(A - \lambda I_n)$ and is a statement about a subspace. They are produced by different computations from different objects, and there is no reason on the face of it that they should be related at all.

They are related, in one direction only. Every eigenvalue has at least one eigenvector, so $\gamma_A(\lambda) \ge 1$ . And a similarity argument shows that a $g$ -dimensional eigenspace forces at least $g$ factors of $(x - \lambda)$ into the characteristic polynomial, giving $\gamma_A(\lambda) \le \alpha_A(\lambda)$ . Since the characteristic polynomial has degree exactly $n$ , the algebraic multiplicities sum to $n$ and none can exceed it. The full chain is $1 \le \gamma_A(\lambda) \le \alpha_A(\lambda) \le n$ .

The practical weight of the inequality is in its strict cases. Summed over all eigenvalues, the algebraic multiplicities always reach $n$ ; the geometric multiplicities reach $n$ only when every eigenvalue is non-deficient, and that is exactly the condition for a basis of eigenvectors to exist. A single eigenvalue with $\gamma_A(\lambda) < \alpha_A(\lambda)$ makes the matrix defective, blocks diagonalisation, and introduces polynomial-in- $t$ terms into the response of any dynamic model built on it.

Definition

Algebraic Multiplicity of an Eigenvalue

AME

Let $A$ be a square matrix with eigenvalue $\lambda$ . The algebraic multiplicity $\alpha_A(\lambda)$ is the highest power of $(x - \lambda)$ that divides the characteristic polynomial $p_A(x) = \det(A - xI_n)$ .

Since $\lambda$ is a root, the factor occurs at least once and $\alpha_A(\lambda) \ge 1$ . Values with $\alpha_A(\lambda) = 1$ are called simple eigenvalues.

Geometric Multiplicity of an Eigenvalue

GME

Let $A$ be a square matrix of size $n$ with eigenvalue $\lambda$ . The geometric multiplicity $\gamma_A(\lambda)$ is the dimension of the eigenspace, $\gamma_A(\lambda) = \dim\left(\mathcal{E}_A(\lambda)\right) = n - \operatorname{rank}\left(A - \lambda I_n\right)$ .

Computed as the count of non-pivot columns in the reduced row-echelon form of $A - \lambda I_n$ .

Defective Eigenvalue

DEF

An eigenvalue $\lambda$ of a square matrix $A$ is defective when $\gamma_A(\lambda) < \alpha_A(\lambda)$ ; the difference $\alpha_A(\lambda) - \gamma_A(\lambda)$ is the defect. A matrix with any defective eigenvalue is itself called defective and admits no basis of eigenvectors.

Concepts

The characteristic polynomial has degree exactly $n$

For an $n \times n$ matrix $A$ , $p_A(x) = \det(A - xI_n)$ is a polynomial of degree $n$ with leading coefficient $(-1)^n$ . The proof is an induction on size, strengthened to cover a more general array: a square matrix whose entries are complex constants or linear polynomials of the form $c - x$ , with at most one such entry in any row and any column. Cofactor expansion about a row containing one of these entries produces a term of degree one higher than the submatrix's, and every other term comes from a submatrix with one fewer $x$ -entry and so contributes lower degree. Since $A - xI_n$ has exactly $n$ entries of the form $c - x$ , one in each row and each column, the result applies directly.

Algebraic multiplicities sum to the size

Factoring $p_A(x)$ completely over $\mathbb{C}$ and grouping equal roots gives $p_A(x) = (-1)^n(x - \lambda_1)^{\alpha_A(\lambda_1)}\cdots(x - \lambda_k)^{\alpha_A(\lambda_k)}$ with distinct $\lambda_i$ . Comparing degrees on the two sides gives $\sum_{i=1}^{k} \alpha_A(\lambda_i) = n$ . This is the counting law that closes an eigenvalue search: once the exponents found sum to $n$ , no eigenvalue remains undiscovered. It also bounds the number of distinct eigenvalues immediately, since each contributes at least $1$ to a sum of $n$ : a square matrix of size $n$ has at most $n$ distinct eigenvalues.

Why the geometric multiplicity cannot exceed the algebraic

Set $g = \gamma_A(\lambda)$ and take a basis $\vec{x}_1, \ldots, \vec{x}_g$ of the eigenspace. Extend it to a basis of $\mathbb{C}^n$ with $n - g$ further vectors and assemble all $n$ vectors as the columns of a matrix $S$ . The columns are independent by construction, so $S$ is non-singular and hence invertible. Because $S^{-1}S = I_n$ , the identity $S^{-1}\vec{x}_i = \vec{e}_i$ holds for $i \le g$ . Computing $S^{-1}AS$ column by column, the first $g$ columns become $\lambda \vec{e}_1, \ldots, \lambda \vec{e}_g$ — zero everywhere except a $\lambda$ on the diagonal.

Completing the argument through similarity

A determinant identity shows that $A$ and $S^{-1}AS$ have the same characteristic polynomial, since $\det(S^{-1})\det(S) = 1$ can be inserted around $\det(A - xI_n)$ and the factors recombined. Expanding $\det\left(S^{-1}AS - xI_n\right)$ down each of the first $g$ columns in turn extracts a factor $(x - \lambda)$ each time, leaving $p_A(x) = (x - \lambda)^{g}\,p_T(x)$ for some square matrix $T$ of size $n - g$ . So $(x - \lambda)$ divides $p_A(x)$ at least $g$ times, which is precisely $\gamma_A(\lambda) \le \alpha_A(\lambda)$ . The final bound $\alpha_A(\lambda) \le n$ follows from the counting law, since multiplicities are positive and they sum to $n$ .

The defect and what it costs

Summing over all distinct eigenvalues gives $\sum_i \gamma_A(\lambda_i) \le \sum_i \alpha_A(\lambda_i) = n$ , with equality precisely when no eigenvalue is defective. The left-hand sum is the total number of independent eigenvectors available, because eigenvectors drawn from distinct eigenspaces are independent. Equality is therefore exactly the condition for a basis of $\mathbb{C}^n$ made of eigenvectors, and hence for diagonalisability. When the sum falls short, the shortfall is the total defect, and it is the number of generalised eigenvectors that must be adjoined to complete a basis.

Where the two multiplicities cannot differ

Two common situations remove the question entirely. If $\alpha_A(\lambda) = 1$ then the inequality $1 \le \gamma_A(\lambda) \le 1$ forces equality, so simple eigenvalues are never defective — a matrix with $n$ distinct eigenvalues is always diagonalisable. If $A$ is Hermitian, or real and symmetric, then a full orthonormal eigenvector basis exists and every eigenvalue satisfies $\gamma_A(\lambda) = \alpha_A(\lambda)$ regardless of repetition. Defectiveness is therefore confined to non-normal matrices with repeated eigenvalues, which is a small but operationally important corner of the subject.

Determining the multiplicities of an eigenvalue

Factor $p_A(x)$ over $\mathbb{C}$ Record the exponent on each distinct linear factor. That exponent is

\alpha_A(\lambda)

Confirm the exponents sum to $n$ If they do not, the factorisation is incomplete. The degree of

p_A

is exactly the size of the matrix.

Is $\alpha_A(\lambda) = 1$ ?If yes,

\gamma_A(\lambda) = 1

is forced by the inequality and no further work is needed for that eigenvalue.

Reduce $A - \lambda I_n$ For each repeated eigenvalue, compute the rank and set

\gamma_A(\lambda) = n - \operatorname{rank}(A - \lambda I_n)

Compare the two counts

\gamma_A(\lambda) = \alpha_A(\lambda)

for every eigenvalue means diagonalisable. Any strict inequality means defective, with defect

\alpha_A(\lambda) - \gamma_A(\lambda)

Total the geometric multiplicitiesThe sum is the number of independent eigenvectors available. Compare against

n

to size the shortfall.

Equations

Degree of the characteristic polynomial

EQ-ME-01

\deg (p_{A} (x)) = \deg (\det (A - x I_{n})) = n

Exactly $n$ entries of $A - xI_n$ contain the indeterminate, one per row and one per column, and each contributes one degree to the leading term.

Complete factorisation and the counting law

EQ-ME-02

p_{A} (x) = (- 1)^{n} \prod_{i}^{=} (x - λ_{i})^{α_{A} (λ_{i})}, \sum_{i}^{=} α_{A} (λ_{i}) = n

Degrees must agree on both sides, so the algebraic multiplicities of the distinct eigenvalues account for exactly $n$ roots.

Multiplicity inequality

EQ-ME-03

1 \leq γ_{A} (λ) \leq α_{A} (λ) \leq n

The central result. Both multiplicities are positive, the geometric never exceeds the algebraic, and neither exceeds the size of the matrix.

Maximum number of distinct eigenvalues

EQ-ME-04

k = \sum_{i}^{=} 1 \leq \sum_{i}^{=} α_{A} (λ_{i}) = n

Each distinct eigenvalue contributes at least one to a total of $n$ , so a matrix of size $n$ has at most $n$ distinct eigenvalues.

Similarity factorisation used in the proof

EQ-ME-05

p_{A} (x) = p_{S^{- 1} A S} (x) = (x - λ)^{g} p_{T} (x), g = γ_{A} (λ)

Extending an eigenspace basis to a basis of $\mathbb{C}^n$ exposes $g$ factors of $(x - \lambda)$ explicitly, which is the mechanism behind $\gamma \le \alpha$ .

Diagonalisability criterion

EQ-ME-06

\sum_{i}^{=} γ_{A} (λ_{i}) = n \Leftrightarrow A has a basis of eigenvectors

The geometric multiplicities sum to $n$ exactly when no eigenvalue is defective, which is exactly when the matrix is diagonalisable.

Defect of a matrix

EQ-ME-07

defect (A) = \sum_{i}^{=} (α_{A} (λ_{i}) - γ_{A} (λ_{i})) = n - \sum_{i}^{=} γ_{A} (λ_{i})

The number of generalised eigenvectors that must be adjoined to the eigenvectors to complete a basis of $\mathbb{C}^n$ .

Variable Definitions

Symbols used on this page
Symbol	Name	Meaning	Domain / type
$A$	Matrix	The square matrix under examination	n x n complex matrix
$n$	Size	Common row and column count; also the degree of $p_A(x)$	positive integer
$k$	Distinct eigenvalue count	Number of distinct roots of the characteristic polynomial	1 to n
$α_{A} (λ)$	Algebraic multiplicity	Exponent of $(x - \lambda)$ in the factored characteristic polynomial	1 to n
$γ_{A} (λ)$	Geometric multiplicity	Dimension of the eigenspace for $\lambda$	1 to alpha
$E_{A} (λ)$	Eigenspace	$\mathcal{N}(A - \lambda I_n)$ , all eigenvectors for $\lambda$ plus $\vec{0}$	subspace of C^n
$S$	Basis matrix	Non-singular matrix whose first $g$ columns are an eigenspace basis	n x n non-singular
$T$	Residual block	Square matrix of size $n - g$ left after extracting the $(x-\lambda)^g$ factor	(n-g) x (n-g) matrix
$p_{A} (x)$	Characteristic polynomial	$\det(A - xI_n)$ , of degree $n$	polynomial over C

Worked Numerical Example

Problem statement

A three-state coupled actuator model has state matrix $D$ . A preliminary analysis found a repeated pole and concluded that the model has two independent slow modes. Determine both multiplicities for every eigenvalue and assess whether that conclusion is justified.

State the matrix
The third state is driven by the first two but does not feed back into them, which shows as a column of zeros above the diagonal:
$D = [\begin{matrix} 1 & 1 & 0 \\ - 1 & 3 & 0 \\ 3 & - 3 & 5 \end{matrix}]$
Expand the determinant down the third column
Only the $(3,3)$ entry of that column is non-zero, so a single cofactor term survives and the $3 \times 3$ determinant collapses to a $2 \times 2$ one.
$p_{D} (x) = (5 - x) \det [\begin{matrix} 1 - x & 1 \\ - 1 & 3 - x \end{matrix}] = (5 - x) [(1 - x) (3 - x) + 1]$
Factor the characteristic polynomial
The bracket expands to $x^2 - 4x + 4$ , a perfect square. Writing the result in monic form gives the algebraic multiplicities directly.
$p_{D} (x) = (5 - x) (x - 2)^{2} = - (x - 5) (x - 2)^{2}$
Record the algebraic multiplicities and check the counting law
The eigenvalues are $\lambda = 2$ with $\alpha_D(2) = 2$ and $\lambda = 5$ with $\alpha_D(5) = 1$ . Their sum is $3 = n$ , as required. The weighted eigenvalue sum $2 + 2 + 5 = 9$ matches $\operatorname{tr}(D) = 1 + 3 + 5$ , and the weighted product $2 \cdot 2 \cdot 5 = 20$ matches $\det(D)$ .
Geometric multiplicity of the simple eigenvalue
For $\lambda = 5$ the algebraic multiplicity is $1$ , so the inequality $1 \le \gamma_D(5) \le 1$ settles the matter without any computation. Reducing $D - 5I_3$ confirms two pivot columns and the eigenvector below.
$E_{D} (5) = span ({[\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]}), γ_{D} (5) = 1 = α_{D} (5)$
Geometric multiplicity of the repeated eigenvalue
This is the case that requires work. Form $D - 2I_3$ and reduce. Rows one and two are identical, but the third row is not a multiple of them, so the rank is $2$ , not $1$ .
$D - 2 I_{3} = [\begin{matrix} - 1 & 1 & 0 \\ - 1 & 1 & 0 \\ 3 & - 3 & 3 \end{matrix}] \to [\begin{matrix} 1 & - 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}]$
Read the eigenspace dimension
The reduced form has pivot columns $1$ and $3$ , leaving one free variable. So $\operatorname{rank}(D - 2I_3) = 2$ and $\gamma_D(2) = 3 - 2 = 1$ , strictly less than $\alpha_D(2) = 2$ .
$E_{D} (2) = span ({[\begin{matrix} 1 \\ 1 \\ 0 \end{matrix}]}), γ_{D} (2) = 1 < 2 = α_{D} (2)$
Total the counts and conclude
The geometric multiplicities sum to $1 + 1 = 2 < 3 = n$ , so there are only two independent eigenvectors in a three-dimensional space. The matrix is defective with defect $1$ , and the preliminary claim of two independent slow modes is wrong: the repeated pole at $\lambda = 2$ supplies exactly one eigenvector, not two.

Result

The model has a defective eigenvalue at $\lambda = 2$ with defect $1$ . Practically this means the matrix cannot be diagonalised and the free response contains a term proportional to $t\,e^{2t}$ alongside the pure exponential. Any simulation or controller tuning that assumed a decomposition into three independent exponential modes will underestimate the transient amplitude, and a generalised eigenvector must be computed to complete the basis.

Applications & Industry Use

Control systems

Repeated poles and transient overshoot

A defective pole of a continuous-time model contributes $t\,e^{\lambda t}$ terms to the free response rather than a pure exponential, producing an initial rise before the eventual decay. Gain schedules tuned against a purely exponential assumption will underpredict peak deviation, so the multiplicity comparison belongs in the model review checklist for every repeated pole.

Structural engineering

Multiplicity of natural frequencies in symmetric structures

Symmetric structures produce repeated natural frequencies whose eigenspaces are genuinely multi-dimensional, so geometric and algebraic multiplicities agree. Any modal extraction that returns fewer independent mode shapes than the repetition count indicates a modelling artefact — typically a non-symmetric damping or follower-force term — rather than a physical property.

Chemical & process engineering

Counting independent conservation laws

The geometric multiplicity of the zero eigenvalue of a stoichiometric or Jacobian matrix is the number of independent conserved quantities in the network. The algebraic multiplicity may be larger, and where it is, the extra factor signals a slowly drifting near-invariant rather than a strict conservation law.

Computing & data

Degenerate spectra in graph analysis

Highly symmetric graphs produce adjacency and Laplacian spectra with large multiplicities. Because these matrices are symmetric, the eigenvalue repetition is always matched by eigenspace dimension, so spectral embedding must select a basis of a subspace rather than a unique set of eigenvectors — a fact that explains why repeated runs can produce rotated but equivalent embeddings.

Aerospace engineering

Mode coalescence at flutter onset

As airspeed approaches the flutter boundary, two structural modes migrate together and the state matrix approaches a defective configuration where two eigenvalues merge but the eigenvectors do not span two dimensions. The near-defect makes the eigenvectors nearly parallel and the eigenvalue sensitivity extremely large, which is the mathematical signature of the instability.

Electrical power systems

Eigenvalue sensitivity near a defect

The sensitivity of an eigenvalue to parameter perturbation is inversely proportional to the inner product of its left and right eigenvectors, which tends to zero as an eigenvalue becomes defective. Near-defective modes therefore move enormous distances under small changes in loading, and identifying them by their multiplicity gap is a prerequisite for robust small-signal assessment.

Design Considerations

Compute the geometric multiplicity for every repeated eigenvalue

The characteristic polynomial cannot distinguish a diagonalisable repeated eigenvalue from a defective one. Any workflow that reports only algebraic multiplicities has left the decisive question unanswered. Add an explicit rank computation of $A - \lambda I_n$ for each repeated root.

Use the inequality to skip work on simple eigenvalues

When $\alpha_A(\lambda) = 1$ , the inequality pins $\gamma_A(\lambda) = 1$ with no computation. For a matrix with all distinct eigenvalues this eliminates the entire multiplicity analysis and guarantees diagonalisability, which is why checking for distinctness first is worth the effort.

Treat defectiveness as structurally fragile

Defective matrices are not generic: an arbitrarily small perturbation of a defective matrix almost always has distinct eigenvalues and is diagonalisable. A model that is exactly defective is usually exactly defective for a structural reason — an idealisation, a symmetry, an exactly repeated component. Where the defect arises from a modelling assumption, test whether it survives realistic parameter variation.

Do not test for defectiveness in floating point alone

Near-defective and exactly defective matrices are numerically indistinguishable: both produce nearly parallel eigenvectors and clustered eigenvalues. The correct instrument is the condition number of the eigenvector matrix, or a Schur or Jordan-structure analysis, rather than an equality test on computed multiplicities.

Exploit symmetry to remove the question

Hermitian and real symmetric matrices are never defective; every eigenvalue satisfies $\gamma_A(\lambda) = \alpha_A(\lambda)$ and an orthonormal eigenvector basis always exists. Where a model can be formulated symmetrically — a stiffness matrix, a covariance, a Gram matrix — doing so removes an entire class of failure at no analytical cost.

Budget for generalised eigenvectors when a defect is real

If the defect is genuine and must be handled, the shortfall $n - \sum_i \gamma_A(\lambda_i)$ is exactly the number of generalised eigenvectors required. These are found by solving $(A - \lambda I_n)\vec{w} = \vec{v}$ for an existing eigenvector $\vec{v}$ , and they complete a basis in which the matrix takes Jordan rather than diagonal form.

Standards & Codes

Notation, interchange and numerical standards that govern how this material is written down, stored and computed in production systems.

Applicable standards, conventions and reference implementations
Reference	Title	Relevance to this topic
`ISO 80000-2`	Quantities and units — Part 2: Mathematics	Governs the Greek symbols $\alpha$ and $\gamma$ used for the two multiplicities and the presentation of the summation and product notation on this page.
`LAPACK reference implementation`	Linear Algebra PACKage driver routines	`xGEESX` and `xGEEVX` return reciprocal condition numbers for eigenvalues and eigenvectors, which are the practical numerical proxy for detecting near-defective behaviour that a multiplicity count cannot see.
`IEEE 754-2019`	IEEE Standard for Floating-Point Arithmetic	Sets the rounding behaviour that makes an exact multiplicity determination impossible in floating point, since a defective double root is perturbed to two distinct roots at the square root of the unit roundoff.
`ISO 7626`	Mechanical vibration and shock — Experimental determination of mechanical mobility	Experimental modal identification under this standard must resolve repeated natural frequencies, where the eigenspace dimension rather than the frequency count determines how many independent mode shapes exist.
`ISO/IEC 40314`	Mathematical Markup Language (MathML) Version 3.0	Encodes the multiplicity inequality and the summation expressions as semantic markup so that the relationships remain machine-readable.

Material Selection

For a mathematical topic, "material" is the numeric representation: the scalar field, storage format and precision the computation is built from.

Representation and precision selection
Representation	Select when	Trade-off
Exact rational arithmetic	Multiplicities must be certified, as in a symbolic model review or a published result.	The only representation in which $\gamma_A(\lambda) < \alpha_A(\lambda)$ can be established with certainty; limited to matrices whose characteristic polynomial factors over the rationals.
Arbitrary-precision floating point	The polynomial does not factor rationally but multiplicities must still be resolved to high confidence.	Allows a repeated root to be separated from a close pair by increasing precision until the behaviour stabilises, at a substantial runtime cost.
IEEE 754 binary64	Routine numerical eigenanalysis of engineering models.	Sufficient for eigenvalues and eigenvectors but incapable of a reliable multiplicity decision; supplement with eigenvector condition numbers.
Schur form rather than eigendecomposition	A matrix is suspected of being defective or near-defective and a stable factorisation is needed.	Always exists and is computed by orthogonal transformations, so it is numerically robust; it gives eigenvalues and invariant subspaces but not eigenvectors directly.
Symmetric or Hermitian storage	The matrix is known self-adjoint, guaranteeing $\gamma = \alpha$ throughout.	Removes defectiveness by construction and halves the computational cost, but silently invalid if the assembly code does not enforce exact symmetry.
Rational canonical or Jordan structure computation	The Jordan block sizes, not just the multiplicities, are required.	Gives the complete similarity invariants including block sizes, but is numerically unstable and viable only in exact arithmetic on modest sizes.

Manufacturing Notes

Implementation notes — how the result is actually produced by hand, by algorithm and by library, including cost and numerical behaviour.

Cost of the multiplicity comparison

Algebraic multiplicities come free with the factorisation of $p_A(x)$ . Each geometric multiplicity costs one row reduction of an $n \times n$ matrix, about $\tfrac{2}{3}n^3$ operations, and is required only for repeated eigenvalues. For a matrix with $r$ repeated eigenvalues the additional work is $\tfrac{2}{3}rn^3$ , which is comparable to a single eigendecomposition when $r$ is small.

Computing the rank reliably

The geometric multiplicity is $n$ minus the rank of $A - \lambda I_n$ , and rank in floating point is a tolerance decision. Determine it from the singular values of $A - \lambda I_n$ with a threshold scaled to $\lVert A \rVert$ and the machine epsilon, not from a count of non-zero pivots in a row reduction, which is far more sensitive to rounding.

Library behaviour

sympy.Matrix.eigenvects returns each eigenvalue with its algebraic multiplicity and a basis of its eigenspace, so the geometric multiplicity is the length of that basis and the comparison is immediate. sympy.Matrix.jordan_form exposes the block structure, which refines the multiplicity data further. NumPy provides neither; numpy.linalg.matrix_rank applied to

A - \lambda I_n

is the closest numerical equivalent.

Detecting near-defectiveness numerically

Compute the eigenvector matrix $S$ and its condition number $\kappa(S)$ . A well-separated diagonalisable matrix gives a modest value; a near-defective one gives a value approaching the reciprocal of the machine epsilon. This single scalar is more informative than any comparison of computed multiplicities and should accompany any reported eigendecomposition of a non-symmetric matrix.

Finding generalised eigenvectors

Where a defect exists, extend the basis by solving $(A - \lambda I_n)\vec{w} = \vec{v}$ with $\vec{v}$ an existing eigenvector for $\lambda$ . The system is consistent precisely when $\vec{v}$ lies in the column space of $A - \lambda I_n$ , which is what the defect guarantees for at least one choice of $\vec{v}$ . Each solve is one row reduction, and the chain continues until the count of vectors for $\lambda$ reaches $\alpha_A(\lambda)$ .

Failure Modes & Common Mistakes

Failure modes, root causes and prevention
Failure mode / mistake	Impact	Root cause	Prevention & detection
Inferring eigenspace dimension from the polynomial	high	Reading $\alpha_A(\lambda)$ off the factorisation and reporting it as the number of independent eigenvectors.	Compute $\gamma_A(\lambda) = n - \operatorname{rank}(A - \lambda I_n)$ for every repeated eigenvalue. The two counts come from different objects.
Assuming $n$ eigenvectors always exist	high	Treating diagonalisability as automatic and building a modal transformation matrix that turns out to be singular.	Verify that the geometric multiplicities sum to $n$ before assembling any eigenvector basis, and check that the assembled matrix is invertible.
Believing $\gamma$ can exceed $\alpha$	medium	Miscounting free variables in the reduced form, or reducing the wrong matrix.	The inequality is a theorem; a computed $\gamma_A(\lambda) > \alpha_A(\lambda)$ is always an arithmetic error. Re-derive both counts.
Declaring a matrix defective from floating-point output	high	Observing two nearly equal computed eigenvalues and nearly parallel eigenvectors, and concluding an exact defect.	Base the claim on exact arithmetic or on structural reasoning. Report eigenvector condition numbers as evidence of near-defectiveness instead.
Stopping the factorisation early	medium	Finding $k$ roots and not confirming that the algebraic multiplicities sum to $n$ .	Divide out each factor found and continue until the quotient is constant. The degree of $p_A$ is exactly $n$ , so the sum is a complete audit.
Confusing defect with rank deficiency	medium	Equating a defective eigenvalue with a singular matrix, or assuming defect implies $\det(A) = 0$ .	Defectiveness concerns the structure of an eigenspace at any eigenvalue; singularity concerns the eigenvalue zero specifically. A non-singular matrix can be defective, and a singular matrix can be diagonalisable.
Assuming a symmetric matrix can be defective	low	Applying general defective-case handling to a Hermitian or real symmetric matrix.	Self-adjoint matrices always satisfy $\gamma = \alpha$ . Exploit the guarantee rather than testing for it, and use the symmetric eigensolver.
Counting geometric multiplicities across different eigenvalues	low	Adding eigenvector counts from distinct eigenvalues into a single eigenspace dimension.	Each eigenvalue owns its own eigenspace. The sum of the geometric multiplicities measures the total eigenvector supply, not the dimension of any one subspace.

FAQs

Why can the geometric multiplicity never exceed the algebraic multiplicity?

Because a $g$ -dimensional eigenspace can be extended to a basis of $\mathbb{C}^n$ , and the resulting similarity transformation puts $g$ columns of the transformed matrix into the form $\lambda \vec{e}_i$ . Expanding the determinant down each of those columns extracts $g$ factors of $(x - \lambda)$ from the characteristic polynomial, so the algebraic multiplicity is at least $g$ . Similar matrices share a characteristic polynomial, which is what transfers the conclusion back to $A$ .

What does it mean in practice for a matrix to be defective?

It means the matrix has fewer than $n$ independent eigenvectors, so no basis of eigenvectors exists and the matrix cannot be diagonalised. For a dynamic model this shows up as response terms of the form $t^{j} e^{\lambda t}$ rather than pure exponentials, and computationally it means the eigenvector matrix is singular and any modal decomposition built on it fails.

Can a matrix with all distinct eigenvalues be defective?

No. Distinct eigenvalues all have algebraic multiplicity one, and the inequality $1 \le \gamma_A(\lambda) \le \alpha_A(\lambda) = 1$ forces geometric multiplicity one as well. Such a matrix always has $n$ independent eigenvectors and is always diagonalisable, which is why checking for distinctness is the fastest route to a diagonalisability decision.

How is the number of distinct eigenvalues bounded?

By the size of the matrix. Each distinct eigenvalue contributes an algebraic multiplicity of at least one, and those multiplicities sum to exactly $n$ because the characteristic polynomial has degree $n$ . So the count of distinct eigenvalues is at most $n$ , with equality precisely when every eigenvalue is simple.

Why is the characteristic polynomial guaranteed to have degree exactly

n

Because $A - xI_n$ contains exactly $n$ entries involving $x$ , one in each row and one in each column, and each contributes one degree to the highest-order term of the determinant expansion. Every other product in the expansion misses at least one diagonal entry and therefore contributes lower degree. The leading coefficient is $(-1)^n$ .

Can I detect defectiveness from a numerical eigensolver?

Not reliably. In floating point a defective matrix presents as clustered eigenvalues with nearly parallel eigenvectors, exactly as a near-defective but diagonalisable matrix would. The practical diagnostic is the condition number of the eigenvector matrix, which grows without bound as a defect is approached, or an eigenvalue condition estimate from a driver such as xGEEVX.

If a matrix is defective, what completes the basis?

Generalised eigenvectors. For an eigenvalue with defect $d$ , one solves $(A - \lambda I_n)\vec{w} = \vec{v}$ starting from an existing eigenvector $\vec{v}$ and continues the chain until $\alpha_A(\lambda)$ vectors are associated with $\lambda$ . In the resulting basis the matrix takes Jordan form: diagonal blocks with $\lambda$ on the diagonal and ones on the superdiagonal.

References

Beezer, R. A. A First Course in Linear Algebra, Version 0.70. University of Puget Sound, 2006. Section PEE, Subsection ME. Licensed under the GNU Free Documentation License v1.2.
Horn, R. A. and Johnson, C. R. Matrix Analysis, 2nd edition. Cambridge University Press, 2012. Section 1.4 and Chapter 3, Canonical Forms.
Golub, G. H. and Van Loan, C. F. Matrix Computations, 4th edition. Johns Hopkins University Press, 2013. Section 7.1, Properties and Decompositions.
ISO 80000-2:2019, Quantities and units — Part 2: Mathematics. International Organization for Standardization.
Anderson, E. et al. LAPACK Users' Guide, 3rd edition. Society for Industrial and Applied Mathematics, 1999.

AI Suggested Questions

Construct a $4 \times 4$ matrix with a single eigenvalue of algebraic multiplicity four and geometric multiplicity two, and show its Jordan structure.
Explain why an arbitrarily small perturbation of a defective matrix is almost always diagonalisable, and what that implies for numerical analysis.
How does the defect of an eigenvalue relate to the sizes of its Jordan blocks?
Derive the sensitivity of a simple eigenvalue to matrix perturbation in terms of its left and right eigenvectors, and show what happens as a defect is approached.
Give an example of a singular matrix that is diagonalisable and a non-singular matrix that is defective.
Show how to compute a generalised eigenvector for a defective eigenvalue and verify that it completes a basis.

Related Calculators

Multiplicity Comparison Tool

Report $\alpha_A(\lambda)$ and $\gamma_A(\lambda)$ for every eigenvalue and flag the defect on each.

Eigenspace Basis Calculator

Reduce $A - \lambda I_n$ and return a basis of the eigenspace with its dimension.

Diagonalisability Checker

Decide whether a matrix admits a basis of eigenvectors and, if not, report the total defect.