Computing Eigenvectors of Linear Transformations

Overview

A linear transformation from a vector space to itself can stretch some directions without rotating them. Those invariant directions are the eigenvectors, and the stretch factors are the eigenvalues. Stated for an abstract space the definition is identical to the matrix case — a non-zero $v \in V$ with $T(v) = \lambda v$ — but the objects involved may be polynomials, matrices, functions or anything else the space contains, so the familiar apparatus of determinants and null spaces cannot be applied directly.

The bridge is coordinatisation. Fix any basis $B$ of $V$ with $n$ elements. The vector representation map $\rho_B \colon V \to \mathbb{C}^n$ is an invertible linear transformation, and relative to $B$ the transformation $T$ acquires an $n \times n$ matrix representation $M^{T}_{B,B}$ . The central result of this page is that $v$ is an eigenvector of $T$ for $\lambda$ if and only if $\rho_B(v)$ is an eigenvector of $M^{T}_{B,B}$ for the same $\lambda$ . The abstract eigenproblem and the concrete matrix eigenproblem are the same problem, wearing different clothes.

That equivalence yields a procedure with no free choices left in it beyond the initial basis: build one representation, extract its characteristic polynomial, find the roots, compute the null space of $M - \lambda I_n$ for each root, and push each resulting column vector back into $V$ through $\rho_B^{-1}$ . The eigenvalues obtained are the eigenvalues of $T$ ; the pulled-back vectors form a basis of the corresponding eigenspace of $T$ .

The result also settles a question of well-definedness that would otherwise be fatal. Two different bases produce two different matrices, so it is not obvious that the answer means anything. It does, because those two matrices are similar via a change-of-basis matrix, and similar matrices share a characteristic polynomial. Eigenvalues, the characteristic polynomial, algebraic multiplicity and geometric multiplicity are therefore intrinsic properties of $T$ , not artefacts of the coordinates used to compute them. Only the particular basis reported for each eigenspace depends on the route taken.

Definition

Eigenvalue and Eigenvector of a Linear Transformation

EELT

Let $T \colon V \to V$ be a linear transformation on a vector space $V$ . A non-zero vector $v \in V$ is an eigenvector of $T$ for the eigenvalue $\lambda$ when $T(v) = \lambda v$ . The scalar $\lambda$ may be zero; the vector may not.

The domain and codomain must be the same space. If $T \colon U \to V$ with $U \ne V$ , the equation $T(v) = \lambda v$ compares elements of two different spaces and has no meaning.

Eigenspace of a Linear Transformation

EELT-SP

For an eigenvalue $\lambda$ of $T \colon V \to V$ , the eigenspace $\mathcal{E}_T(\lambda)$ is the set of all eigenvectors of $T$ for $\lambda$ together with the zero vector of $V$ . Equivalently $\mathcal{E}_T(\lambda) = \mathcal{K}(T - \lambda I_V)$ , the kernel of the transformation $v \mapsto T(v) - \lambda v$ , which makes it a subspace of $V$ .

Adjoining $\vec{0}$ is what converts a bare collection of vectors into a subspace; the zero vector is never itself an eigenvector.

Algebraic and Geometric Multiplicity for a Transformation

AME-GME

Let $M$ be any matrix representation of $T$ relative to a single basis used for both domain and codomain. The algebraic multiplicity $\alpha_T(\lambda)$ is the highest power of $(x - \lambda)$ dividing the characteristic polynomial of $M$ . The geometric multiplicity $\gamma_T(\lambda) = \dim(\mathcal{E}_T(\lambda))$ . Both are well defined because every such representation is similar to every other.

Always $1 \le \gamma_T(\lambda) \le \alpha_T(\lambda)$ . Equality for every eigenvalue is exactly the condition under which $T$ admits a diagonal representation.

Concepts

Representation converts the eigenproblem, faithfully

The governing result states that for a basis $B$ of $V$ and a linear transformation $T \colon V \to V$ , a vector $v \in V$ is an eigenvector of $T$ for $\lambda$ precisely when $\rho_B(v)$ is an eigenvector of $M^{T}_{B,B}$ for $\lambda$ . The forward direction follows in three lines: $M^{T}_{B,B}\,\rho_B(v) = \rho_B(T(v)) = \rho_B(\lambda v) = \lambda\, \rho_B(v)$ , using in turn the defining property of a matrix representation, the eigenvector hypothesis, and the linearity of $\rho_B$ . The reverse direction runs the same chain backwards through $\rho_B^{-1}$ , which exists because $\rho_B$ is an isomorphism.

Why the answer does not depend on the basis

Choosing a second basis $C$ of $V$ produces a second matrix $M^{T}_{C,C}$ , and the two are related by the change-of-basis matrix through $M^{T}_{B,B} = C_{B,C}^{-1} M^{T}_{C,C} C_{B,C}$ . That is precisely the statement that the two representations are similar. Similar matrices have identical characteristic polynomials, hence identical eigenvalues with identical algebraic multiplicities, and their eigenspaces have equal dimensions. The eigenvalues of $T$ are therefore an intrinsic invariant, and it is legitimate to speak of the characteristic polynomial of a linear transformation.

Eigenspaces are the same subspace, not the same list

Basis independence applies to the eigenspace as a set, not to the particular spanning vectors that a computation happens to emit. Two bases $B$ and $C$ will generally return two different-looking bases for $\mathcal{E}_T(\lambda)$ when $\gamma_T(\lambda) \ge 2$ , because the null space of $M - \lambda I_n$ is computed in different coordinates. The two answers span the same subspace of $V$ . Confirming that requires the standard argument: show each reported vector lies in the span of the other set, and note that the two sets have equal size, so equal dimension forces set equality.

Choosing a basis that makes the arithmetic cheap

No theorem constrains the choice of $B$ , so it should be chosen to make the representation easy to build by hand. For $P_n$ , the monomial basis $\{1, x, x^2, \ldots, x^n\}$ reads coefficients straight off. For $M_{mn}$ , the basis of matrices with a single $1$ entry does the same. The pay-off is that $\rho_B$ and $\rho_B^{-1}$ become transcription rather than computation, and the only real work is the matrix eigenproblem. A poorly chosen basis does not change the answer, only the labour required to reach it.

The diagonalisable case and its meaning

If the representation is diagonalisable, then $V$ has a basis $C$ consisting entirely of eigenvectors of $T$ , obtained by un-coordinatising a full set of $n$ independent eigenvectors of the matrix. Representing $T$ relative to that basis gives a diagonal matrix whose diagonal entries are the eigenvalues, repeated according to multiplicity. Read in reverse, matrix diagonalisation is the search for a basis in which a linear transformation acts by independent scaling along $n$ directions — decoupling a coupled operator into $n$ scalar operators.

Complex scalars are not optional

The characteristic polynomial of an $n \times n$ matrix has exactly $n$ roots counted with multiplicity over $\mathbb{C}$ , and may have none at all over $\mathbb{R}$ . A rotation-like transformation on a real space has no real eigenvalue, yet has a complete set of complex ones. Working over $\mathbb{C}$ therefore guarantees that eigenvalues exist, and complex eigenvectors of a real transformation arrive in conjugate pairs. Un-coordinatising a complex eigenvector yields an element of $V$ with complex coefficients — a polynomial with complex coefficients, or a complex matrix — which is legitimate provided $V$ was taken over $\mathbb{C}$ from the outset.

Procedure: eigenvalues and eigenvectors of $T \colon V \to V$

Choose a convenient basis $B$ Any basis of

V

works. Pick one whose coordinates can be read off by inspection, such as the monomial basis for polynomials.

Build the representation $M = M^{T}_{B,B}$ Column

j

\rho_B(T(u_j))

for the

j

-th basis vector

u_j

. Use

B

for both domain and codomain.

Factor the characteristic polynomialCompute

p(x) = \det(M - x I_n)

and find its roots over

\mathbb{C}

. These roots are the eigenvalues of

T

; the exponent of

(x - \lambda)

\alpha_T(\lambda)

Solve for each eigenspace in coordinatesFor each

\lambda

, compute a basis of the null space

\mathcal{N}(M - \lambda I_n)

by row reduction. The number of vectors is

\gamma_T(\lambda)

Un-coordinatise with $\rho_B^{-1}$ Map each null-space basis vector back into

V

by reading its entries as coefficients on the elements of

B

. The images form a basis of

\mathcal{E}_T(\lambda)

Check multiplicitiesIf

\gamma_T(\lambda) = \alpha_T(\lambda)

for every eigenvalue, the eigenvectors form a basis of

V

and

T

has a diagonal representation. Otherwise it does not.

Equations

Eigenvector condition for a transformation

EQ-CELT-01

T (v) = λ v, v \neq \vec{0}, v \in V

The defining relation. The exclusion of the zero vector is essential: without it every scalar would be an eigenvalue of every transformation.

Representation transports the eigenproblem

EQ-CELT-02

T (v) = λ v \Leftrightarrow M_{B}^{,} ρ_{B} (v) = λ ρ_{B} (v)

The equivalence that licenses the whole procedure. It holds for every basis $B$ of $V$ , with no restriction whatever on the choice.

Characteristic polynomial of the transformation

EQ-CELT-03

p_{T} (x) = \det (M_{B}^{,} - x I_{n})

Independent of $B$ , because any two representations relative to a single basis are similar and similar matrices have equal characteristic polynomials.

Similarity of two representations

EQ-CELT-04

M_{B}^{,} = C_{B}^{,} M_{C}^{,} C_{B, C}

Here $C_{B,C}$ is the change-of-basis matrix from $B$ to $C$ , itself the representation of the identity transformation on $V$ .

Eigenspace recovered by un-coordinatisation

EQ-CELT-05

Eigenspace recovered by un-coordinatisation

The image of a subspace of $\mathbb{C}^n$ under an isomorphism is a subspace of $V$ of the same dimension, so a basis maps to a basis.

Multiplicity bounds

EQ-CELT-06

1 \leq γ_{T} (λ) \leq α_{T} (λ), \sum_{λ} α_{T} (λ) = n

Algebraic multiplicities always exhaust the dimension of $V$ over $\mathbb{C}$ . Geometric multiplicities do so only when $T$ is diagonalisable.

Diagonal representation on an eigenbasis

EQ-CELT-07

M_{C}^{,} = [\begin{matrix} λ_{1} & 0 & \dots & 0 \\ 0 & λ_{2} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & λ_{n} \end{matrix}]

Obtained when $C$ is a basis of $V$ made entirely of eigenvectors of $T$ . Each basis vector is scaled and none is mixed with any other.

Variable Definitions

Symbols used on this page
Symbol	Name	Meaning	Domain / type
$T$	Linear transformation	A linear map from $V$ to itself, the object whose eigenstructure is sought	linear transformation V to V
$V$	Vector space	The common domain and codomain, of finite dimension $n$	vector space over C
$B$	Computation basis	Any basis of $V$ , chosen for convenience; used for both domain and codomain	basis of V, size n
$ρ_{B}$	Vector representation map	The isomorphism sending $v \in V$ to its coordinate column relative to $B$	isomorphism V to C^n
$M_{B}^{,}$	Matrix representation	The $n \times n$ matrix whose $j$ -th column is $\rho_B(T(u_j))$	n x n complex matrix
$λ$	Eigenvalue	Scale factor satisfying $T(v) = \lambda v$ for some non-zero $v$	complex scalar
$E_{T} (λ)$	Eigenspace	Subspace of $V$ containing all eigenvectors for $\lambda$ , plus $\vec{0}$	subspace of V
$α_{T} (λ)$	Algebraic multiplicity	Multiplicity of $\lambda$ as a root of the characteristic polynomial	integer 1 to n
$γ_{T} (λ)$	Geometric multiplicity	Dimension of the eigenspace $\mathcal{E}_T(\lambda)$	integer 1 to alpha
$Change-of-basis matrix$	Change-of-basis matrix	Representation of the identity on $V$ relative to $B$ and $C$	n x n nonsingular matrix

Worked Numerical Example

Problem statement

A signal-conditioning stage acts on quadratic calibration curves. Modelled on the space $P_2$ of polynomials of degree at most two, the stage is the linear transformation $T(a + bx + cx^2) = (2a + b) + (a + 2b)x + (a - b + 3c)x^2$ . Find the calibration curves that pass through the stage unchanged up to a gain, and the gains themselves.

Choose the monomial basis
Take $B = \{1,\ x,\ x^2\}$ , so $n = \dim(P_2) = 3$ . Coordinates relative to $B$ are just the coefficient triples, which makes $\rho_B$ and $\rho_B^{-1}$ pure transcription.
$ρ_{B} (a + b x + c x^{2}) = [\begin{matrix} a \\ b \\ c \end{matrix}]$
Build the matrix representation
Apply $T$ to each basis element and record the coordinate column. $T(1) = 2 + x + x^2$ , $T(x) = 1 + 2x - x^2$ , and $T(x^2) = 3x^2$ . These three columns assemble into $M = M^{T}_{B,B}$ .
$M = [\begin{matrix} 2 & 1 & 0 \\ 1 & 2 & 0 \\ 1 & - 1 & 3 \end{matrix}]$
Factor the characteristic polynomial
The third column of $M - x I_3$ is $(0, 0, 3-x)$ , so expansion along that column is immediate. The eigenvalues are $\lambda = 3$ with algebraic multiplicity $2$ , and $\lambda = 1$ with algebraic multiplicity $1$ .
$p_{T} (x) = (3 - x) [(2 - x)^{2} - 1] = (3 - x) (x - 1) (x - 3) = (3 - x)^{2} (1 - x)$
Solve for the eigenspace at $\lambda = 3$
Row-reduce $M - 3I_3$ . Every row is a multiple of $[\,1\ \ -1\ \ 0\,]$ , so the rank is $1$ and the null space has dimension $2$ : $\gamma_T(3) = 2 = \alpha_T(3)$ .
$Solve for the eigenspace at $\lambda = 3$$
Solve for the eigenspace at $\lambda = 1$
Row-reduce $M - I_3$ . Two pivots appear, so the null space is one-dimensional and $\gamma_T(1) = 1 = \alpha_T(1)$ . Setting the free variable $x_3 = 1$ gives the single basis vector below.
$Solve for the eigenspace at $\lambda = 1$$
Un-coordinatise into $P_2$
Read each column as coefficients on $1$ , $x$ and $x^2$ . The null-space basis $\{(1,1,0), (0,0,1)\}$ for $\lambda = 3$ becomes $\{1 + x,\ x^2\}$ , and $(-1,1,1)$ for $\lambda = 1$ becomes $-1 + x + x^2$ .
$Un-coordinatise into $P_2$$
Verify directly in $P_2$
Substitute back into the original rule, never into the matrix. For $1 + x$ take $a = 1$ , $b = 1$ , $c = 0$ : $T(1+x) = 3 + 3x = 3(1+x)$ . For $-1 + x + x^2$ take $a = -1$ , $b = 1$ , $c = 1$ : $T(-1+x+x^2) = -1 + x + x^2$ , an eigenvalue of $1$ .
$Verify directly in $P_2$$
Confirm basis independence with a second basis
Repeat with $D = \{1,\ 1+x,\ 1+x+x^2\}$ . The representation becomes $N = M^{T}_{D,D}$ below, whose characteristic polynomial is again $(3-x)^2(1-x)$ . Its eigenspace at $\lambda = 3$ un-coordinatises to $\{1+x,\ 1+x+x^2\}$ — a different list, but $1+x+x^2 = (1+x) + x^2$ , so it spans the same subspace. Its eigenvector for $\lambda = 1$ un-coordinatises to exactly $-1 + x + x^2$ .
$N = [\begin{matrix} 1 & 0 & 0 \\ 0 & 3 & 0 \\ 1 & 0 & 3 \end{matrix}], M = S N S^{- 1}, S = [\begin{matrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix}]$

Result

The stage has two gains: it multiplies any curve in the two-dimensional family spanned by $1 + x$ and $x^2$ by exactly $3$ , and multiplies the single curve $-1 + x + x^2$ by $1$ , leaving it untouched. Because $\gamma_T(\lambda) = \alpha_T(\lambda)$ for both eigenvalues, the three eigenpolynomials form a basis of $P_2$ : every quadratic calibration curve decomposes uniquely into a part amplified threefold and a part passed through unchanged, and the stage is fully characterised by those two numbers.

Applications & Industry Use

Structural dynamics

Mode shapes as functions, not coordinate lists

Discretised beam and plate models represent deflection by coefficients on a set of shape functions, so the natural state space is a function space rather than $\mathbb{C}^n$ . Modal analysis then asks for eigenvectors of an operator on that space. Coordinatising with the shape-function basis converts the question into a matrix eigenproblem; un-coordinatising returns the mode shapes as actual deflection functions, which is the form an engineer can plot and interpret.

Control systems

Basis-invariant assessment of a plant

State-space models of a single physical plant differ by a similarity transformation whenever the state variables are redefined — currents versus fluxes, positions versus modal amplitudes. Because eigenvalues survive similarity, pole locations, stability margins and time constants can be computed in whichever state basis is most convenient and quoted as properties of the plant itself.

Quantum and molecular modelling

Operators on spaces of basis functions

Electronic-structure calculations express an energy operator in a chosen basis of atomic orbitals. The resulting matrix eigenproblem yields orbital energies and coefficient vectors; the coefficient vectors are then un-coordinatised back into functions to obtain the orbitals themselves. Changing basis set changes the matrix but not the underlying spectrum, which is the property that makes basis-set convergence a meaningful concept.

Signal processing

Filter stages acting on polynomial or spline models

A conditioning stage applied to a curve represented in a polynomial or B-spline basis is a linear transformation of the coefficient space. Its eigenpolynomials are the curve shapes the stage merely rescales; identifying them shows which features of a calibration profile pass through a processing chain undistorted and which are attenuated.

Process engineering

Decoupling interacting unit operations

A recycle loop couples composition variables so that no single stream can be adjusted independently. Representing the loop transformation on a suitable state space and finding its eigenvectors identifies combinations of compositions that evolve independently. Control in those coordinates reduces a multivariable problem to a set of single-loop problems.

Computer graphics and geometry

Invariant directions of a transformation on a shape space

Deformation operators acting on control-point or blend-shape spaces are linear transformations on a space whose elements are geometric configurations rather than triples. Their eigenvectors are configurations that the operator merely scales, which is how principal deformation modes are extracted from a rig and how a large blend-shape basis is reduced to a compact one.

Design Considerations

Confirm domain and codomain coincide before starting

The eigenvector equation compares $T(v)$ with $\lambda v$ , so both must live in the same space. A transformation $T \colon P_4 \to P_5$ has no eigenvalues at all, however well behaved it is. This check costs nothing and rules out an entire class of meaningless computations.

Spend the effort on choosing the basis, not on the algebra

The representation must be built by hand at least once, and a basis in which $\rho_B$ is transcription rather than a linear solve saves the bulk of the work. Monomials for polynomial spaces and single-entry matrices for matrix spaces are almost always the right first choice; specialised bases become attractive only when they make $T$ visibly block-structured.

Report eigenspaces, not eigenvectors

When a geometric multiplicity exceeds one, the individual basis vectors returned by a null-space computation are arbitrary. Two correct computations can disagree vector by vector while agreeing perfectly as subspaces. Documentation and test cases should therefore state the eigenspace and its dimension, and any comparison between two results must be a subspace comparison.

Treat the multiplicity gap as the headline diagnostic

The single most informative output beyond the eigenvalues is whether $\gamma_T(\lambda) = \alpha_T(\lambda)$ for every $\lambda$ . Equality throughout means a basis of eigenvectors exists and the transformation decouples completely. A deficiency means it does not, and that any attempt to build a diagonal representation will fail no matter how the search is organised.

Verify against the original rule, never the representation

Substituting a candidate eigenvector back into $M$ only re-checks the null-space arithmetic. Substituting the un-coordinatised element into the original definition of $T$ additionally checks the representation build and both coordinate maps, which is where hand errors actually occur.

Decide early whether complex scalars are admissible

If $V$ is a real vector space, a real transformation may have no eigenvectors whatever. Widening the scalars to $\mathbb{C}$ restores existence but produces eigenvectors with complex coefficients, which may or may not be physically interpretable in the model at hand. Where complex eigenvalues arise in conjugate pairs, the real span of the real and imaginary parts of a complex eigenvector gives a two-dimensional invariant subspace that stays inside the real space.

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	Fixes the typographic conventions used for the objects on this page: italic for the transformation symbol $T$ and scalars such as $\lambda$ , upright for operators such as $\det$ and $\dim$ , and the notation for mappings between sets.
`LAPACK / BLAS reference`	Linear Algebra PACKage reference implementation	Once the abstract problem has been coordinatised, the remaining work is a dense eigenproblem handled by `xGEEV` for the general case and `xSYEV` or `xHEEV` when the representation is symmetric or Hermitian.
`IEEE 754-2019`	IEEE Standard for Floating-Point Arithmetic	Defines the arithmetic in which the characteristic polynomial roots and null spaces are computed numerically, and hence the rounding behaviour that makes an exactly repeated eigenvalue impossible to certify from floating-point output alone.
`ISO/IEC 40314`	Mathematical Markup Language (MathML) Version 3.0	Encodes the subscripted representation symbols $M^{T}_{B,B}$ , $\rho_B$ and $C_{B,C}$ semantically, so that a screen reader distinguishes the transformation superscript from an exponent.
`W3C WCAG 2.1 AA`	Web Content Accessibility Guidelines	Matrices and coordinate columns on this page are semantic markup with alternative text rather than images, keeping the notation legible under magnification and available to assistive technology.

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 on the representation	The transformation has integer or rational coefficients and the multiplicity structure is the deliverable.	Gives certified algebraic and geometric multiplicities, but characteristic-polynomial roots are only exact when the polynomial factors over the rationals.
Symbolic algebra over an algebraic extension	Eigenvalues are irrational or complex and exact eigenvectors are required, for example to prove a decoupling result.	Exact throughout, at the cost of expression swell that makes anything beyond about size ten impractical.
IEEE 754 binary64 with a general eigensolver	The representation comes from measured or simulated data and approximate eigenvalues suffice.	Fast and robust, but repeated eigenvalues split into nearby distinct ones, so geometric multiplicity cannot be read from the output without a tolerance.
Complex scalars throughout	The transformation is not known in advance to have real spectrum, or oscillatory behaviour is expected.	Guarantees a full complement of eigenvalues and doubles storage; interpreting a complex eigenvector as a physical configuration requires an extra modelling step.
Real arithmetic with a real Schur form	The space is genuinely real and complex coefficients would be meaningless in the model.	Keeps every quantity real by using two-dimensional invariant blocks in place of complex conjugate pairs, but no longer delivers eigenvectors in the literal sense.
Dense storage of the representation	The representation is built from an abstract rule, as here, so it is dense by construction.	Sparse storage rarely helps, because a matrix representation of an operator on a polynomial or matrix space is usually full; the size of the space, not its sparsity, is the limiting factor.

Manufacturing Notes

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

Cost breakdown of the procedure

Building the representation costs $n$ applications of $T$ plus $n$ coordinate solves; with a well-chosen basis the latter are free. The eigenproblem itself dominates at roughly $25n^3$ floating-point operations for a general dense matrix when eigenvectors are wanted. Un-coordinatisation is $O(n^2)$ at most. In short, the abstract wrapper is cheap and the matrix core is the whole cost.

Building the representation reliably by hand

Work one basis vector at a time and write each image as an explicit linear combination of the basis before extracting the column. Recording the intermediate combination is what catches sign errors. A frequent slip is to write the coordinate vectors as rows; the definition places $\rho_B(T(u_j))$ in column $j$ , and transposing the result silently replaces $T$ by a different transformation.

Library behaviour and the missing abstraction

Numerical libraries expose eigensolvers for matrices only — numpy.linalg.eig, scipy.linalg.eig, MATLAB's eig, Eigen's EigenSolver. None of them knows about abstract vector spaces, so the coordinatisation and un-coordinatisation steps are the caller's responsibility. Symbolic systems fare better: SymPy's polynomial and matrix classes make the representation build scriptable, and its eigenvects returns exact eigenspace bases with multiplicities attached.

Certifying multiplicities in floating point

A repeated eigenvalue is structurally unstable under rounding: a perturbation of size $\varepsilon$ can split an eigenvalue of algebraic multiplicity $k$ into $k$ values differing by order $\varepsilon^{1/k}$ . Clustering the computed spectrum with a tolerance is the practical remedy, and the corresponding invariant subspace should be extracted as a whole rather than as individual eigenvectors.

Two independent verification routes

First, substitute each un-coordinatised eigenvector into the original rule for $T$ and confirm the scaling exactly. Second, recompute the representation relative to a second basis and confirm the characteristic polynomial is unchanged; agreement here validates the change-of-basis reasoning as well as the arithmetic, and disagreement localises the error to the representation build.

Failure Modes & Common Mistakes

Failure modes, root causes and prevention
Failure mode / mistake	Impact	Root cause	Prevention & detection
Applying the method when the codomain differs from the domain	high	Treating any linear transformation as an eigenproblem candidate without checking that $T$ maps $V$ to $V$ .	Verify $T \colon V \to V$ first. For $T \colon U \to V$ with $U \ne V$ , eigenvalues are undefined and the correct questions are about rank, kernel and range.
Reporting coordinate columns as the eigenvectors	high	Stopping after the matrix eigenproblem and omitting the $\rho_B^{-1}$ step.	The answer must be an element of $V$ — a polynomial, a matrix, whatever $V$ contains. A column of numbers is a representation of the answer, not the answer.
Using different bases for domain and codomain	high	Building $M^{T}_{B,C}$ with $B \ne C$ out of habit from general representation work.	A rectangular or mixed-basis representation has no meaningful eigenvalues. Use one basis in both roles, giving a square matrix similar to every other such representation.
Concluding two eigenspaces differ because their bases differ	medium	Comparing the vectors returned by two computations elementwise instead of comparing the subspaces they span.	Show mutual containment, or confirm equal dimension plus one-way containment. Different bases of the same subspace are the expected outcome, not an error.
Assuming every eigenvalue yields as many eigenvectors as its multiplicity	medium	Conflating algebraic with geometric multiplicity, which coincide only for diagonalisable transformations.	Compute $\gamma_T(\lambda)$ as the nullity of $M - \lambda I_n$ independently, and compare it with the exponent in the characteristic polynomial.
Searching for a real eigenvalue that does not exist	medium	Restricting to real scalars for a transformation with rotational character, whose characteristic polynomial has no real roots.	Work over $\mathbb{C}$ so that $n$ roots always exist, and interpret conjugate pairs as two-dimensional invariant subspaces if the model must remain real.
Transposing the representation during construction	medium	Writing each $\rho_B(T(u_j))$ as a row of the matrix rather than a column.	Check one column against its defining basis vector before proceeding. The transpose generally has the same eigenvalues but different eigenvectors, so the error is not self-announcing.
Trusting a floating-point eigensolver on an exactly repeated eigenvalue	medium	Reading distinct computed values as distinct eigenvalues when they are a split multiple root.	Cluster the spectrum with a tolerance scaled to the matrix norm, and extract the invariant subspace for a cluster rather than individual eigenvectors.
Including the zero vector as an eigenvector	low	Reading the eigenspace definition as if $\vec{0}$ qualified, since $T(\vec{0}) = \lambda \vec{0}$ holds for every $\lambda$ .	Keep the definitions separate: the eigenspace contains $\vec{0}$ so that it is a subspace, but $\vec{0}$ is never an eigenvector.

FAQs

Does the choice of basis change the eigenvalues I compute?

No. Two representations of the same transformation relative to two bases are similar matrices, and similar matrices have identical characteristic polynomials. The eigenvalues, their algebraic multiplicities and the dimensions of the eigenspaces are therefore properties of the transformation alone. Only the coordinate columns and the particular basis reported for an eigenspace change.

Why do two bases give me visibly different eigenvectors for the same eigenvalue?

Because a null-space computation returns one basis of a subspace, and a subspace of dimension two or more has infinitely many bases. Both answers are correct and span the same eigenspace of $V$ . Confirm this by expressing each reported vector as a linear combination of the other set, then invoke equal dimension to conclude the subspaces coincide.

Can a linear transformation between different spaces have eigenvalues?

No. The equation $T(v) = \lambda v$ requires $T(v)$ and $v$ to be comparable, which forces the codomain to be the domain. For a transformation between genuinely different spaces the meaningful invariants are the rank, the kernel and the range, and the analogue of an eigen-decomposition is the singular value decomposition of a representation.

What does it mean if the geometric multiplicity is smaller than the algebraic multiplicity?

It means the transformation cannot be diagonalised: there are not enough independent eigenvectors to form a basis of $V$ , so no choice of basis makes the representation diagonal. Physically it signals a defective mode, and dynamically it produces responses with polynomial-in-time factors alongside the exponential ones rather than pure exponentials.

How do I know I have found every eigenvalue?

Count algebraic multiplicities. Over $\mathbb{C}$ the characteristic polynomial has degree $n = \dim(V)$ and factors completely, so the multiplicities must sum to $n$ . If they do not, a root has been missed or mis-multiplied. Over the reals no such guarantee exists, which is one reason the theory is set in $\mathbb{C}$ .

Is there any way to avoid building a matrix representation at all?

For small or highly structured transformations, guessing and verifying works: propose a candidate, apply $T$ , and see whether the result is a scalar multiple. That confirms eigenvectors but never proves the list is complete. Establishing completeness requires the characteristic polynomial, and that requires a representation.

Which basis should I pick when the transformation is defined on a space of matrices?

Start with the basis of matrices having a single entry equal to one and the rest zero, ordered consistently. Coordinates are then just the entries listed in that order, so both coordinate maps become transcription. Only if $T$ visibly preserves a structural subspace, such as the symmetric matrices, is a specialised basis worth the extra setup.

References

Beezer, R. A. A First Course in Linear Algebra, Version 0.70. University of Puget Sound, 2006. Section CB, Subsection CELT. Licensed under the GNU Free Documentation License v1.2.
ISO 80000-2:2019, Quantities and units — Part 2: Mathematics. International Organization for Standardization.
Golub, G. H. and Van Loan, C. F. Matrix Computations, 4th edition. Johns Hopkins University Press, 2013. Chapter 7, the unsymmetric eigenvalue problem.
Anderson, E. et al. LAPACK Users' Guide, 3rd edition. Society for Industrial and Applied Mathematics, 1999.
Halmos, P. R. Finite-Dimensional Vector Spaces. Springer, 1974. Treatment of linear transformations and their invariant subspaces without reference to coordinates.

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