Eigenvalues and Eigenvectors of Linear Transformations

Overview

Most of what a linear transformation does to a vector is rotate it, stretch it, and mix it with other directions. Eigenvectors are the exceptional inputs on which the mixing disappears entirely: the output points along the same line as the input, scaled by a single number. That number is the eigenvalue. Because the condition is expressed purely in terms of the transformation and the vector space operations, it requires no basis, no coordinates and no matrix, and it therefore applies verbatim to transformations of abstract spaces.

The restriction that makes the definition sensible is that the transformation must map a space to itself. Comparing $T(\vec{v})$ with $\vec{v}$ requires them to live in the same space, so eigenvalues are defined for $T\colon V \to V$ and are meaningless for a transformation between genuinely different spaces. This is the same restriction that makes a matrix representation square when one basis is used at both ends, and the two facts are aspects of a single situation.

Eigenvalues carry the behaviour that survives a change of description. A transformation may be represented by wildly different matrices depending on the basis chosen, but the set of eigenvalues is fixed, because an eigenvector is defined without reference to coordinates. This is what makes eigenvalues the right vocabulary for stability, resonance, growth rates and principal directions: they are properties of the operator, not of the person who chose the axes.

The practical route to eigenvalues runs through representation. Choose any convenient basis, form the square representation of the transformation relative to that basis at both ends, and compute the eigenvalues of that matrix by the standard techniques. The eigenvectors of the matrix are the coordinate vectors of the eigenvectors of the transformation, so decoding them returns polynomials, matrices or whatever objects the space contains. Choosing a basis made of eigenvectors, when one exists, is what produces a diagonal representation and is the entire content of diagonalisation.

Definition

Eigenvalue and Eigenvector of a Linear Transformation

EELT

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

Excluding the zero vector is essential. The equation $T(\vec{0}) = \lambda\vec{0}$ holds for every scalar, so admitting $\vec{0}$ would make every scalar an eigenvalue and destroy the definition.

Eigenspace of a Transformation

EST

For a fixed scalar $\lambda$ , the eigenspace is the set of all vectors satisfying $T(\vec{v}) = \lambda\vec{v}$ , including the zero vector. Equivalently it is the kernel of the transformation $T - \lambda I_V$ , which makes it a subspace of $V$ . Its dimension is the geometric multiplicity of $\lambda$ , and $\lambda$ is an eigenvalue precisely when that dimension is at least one.

Invariant Subspace

ISS

A subspace $W \subseteq V$ is invariant under $T$ when $T(\vec{w}) \in W$ for every $\vec{w} \in W$ . Every eigenspace is invariant, and every one-dimensional invariant subspace is spanned by an eigenvector. Invariance is the structural reason eigenvectors give block-diagonal and diagonal representations.

Concepts

Why the transformation must map a space to itself

The eigenvector equation compares an output with an input. For $T\colon U \to V$ with $U$ and $V$ genuinely different spaces, $T(\vec{v})$ and $\vec{v}$ cannot be compared and the equation has no meaning. Only for $T\colon V \to V$ is the question well posed. In representation terms, this is why eigenvalue analysis uses a single basis at both ends, producing a square matrix $M^{T}_{B,B}$ , whereas a transformation between different spaces has a rectangular representation with no eigenvalues at all.

Eigenspaces are kernels

The equation $T(\vec{v}) = \lambda\vec{v}$ can be rewritten as $(T - \lambda I_V)(\vec{v}) = \vec{0}$ , so the eigenvectors for $\lambda$ , together with the zero vector, form the kernel of $T - \lambda I_V$ . This reformulation converts an eigenvalue question into a kernel question and explains three facts at once: the eigenspace is a subspace, $\lambda$ is an eigenvalue exactly when $T - \lambda I_V$ fails to be injective, and every technique for computing kernels applies unchanged to computing eigenspaces.

The meaning of a zero eigenvalue

Zero is an eigenvalue of $T$ exactly when there is a non-zero vector with $T(\vec{v}) = \vec{0}$ , that is, exactly when the kernel is non-trivial and $T$ fails to be injective. For a transformation of a finite-dimensional space to itself, that is also exactly when $T$ fails to be invertible. A zero eigenvalue is therefore not a degenerate curiosity but the signature of information loss, and the corresponding eigenspace is the kernel itself.

Eigenvectors are determined only up to scale

If $\vec{v}$ is an eigenvector for $\lambda$ then so is $\alpha\vec{v}$ for any non-zero scalar $\alpha$ , since $T(\alpha\vec{v}) = \alpha T(\vec{v}) = \lambda(\alpha\vec{v})$ . Eigenvectors therefore identify invariant directions, not particular vectors, and any reported eigenvector is a representative of a whole line. When eigenvectors must be compared across computations or stored in a file, adopt a normalisation convention — unit norm with a fixed sign, or a chosen entry set to one — because software will otherwise return arbitrary multiples.

Computation through a representation

Eigenvalues of a transformation are computed by forming any square representation $M^{T}_{B,B}$ and finding the eigenvalues of that matrix. The correspondence is exact: $\vec{v}$ is an eigenvector of $T$ for $\lambda$ if and only if $\rho_B(\vec{v})$ is an eigenvector of $M^{T}_{B,B}$ for the same $\lambda$ . This follows immediately from the fundamental theorem of matrix representation, and it means the choice of basis affects the arithmetic but never the answer. Representations relative to different bases are similar matrices, and similar matrices share a characteristic polynomial.

Existence, scarcity and the underlying field

Over the complex numbers, every linear transformation of a non-zero finite-dimensional space has at least one eigenvalue, because the characteristic polynomial of any representation has a root. Over the real numbers this fails: a rotation of the plane through an angle other than a multiple of $\pi$ maps no direction to itself. Eigenvalues can also be scarce in a different way — differentiation on the space of polynomials of bounded degree has only the eigenvalue zero, with the constants as its eigenspace, because differentiating strictly lowers degree.

Finding eigenvalues of a transformation

Confirm the transformation maps $V$ to $V$ Eigenvalues are undefined otherwise. If the codomain differs from the domain, no eigenvalue question exists.

Try the definition directlyFor structured operators — transposition, scaling, projection, reflection — the equation

T(\vec{v}) = \lambda\vec{v}

can often be solved by inspection, giving eigenvalues with no arithmetic at all.

Otherwise choose a convenient basisPick a basis

B

V

that makes coordinatisation easy, and build the square representation

M^{T}_{B,B}

using

B

at both ends.

Solve the matrix eigenproblemFind the roots of the characteristic polynomial for small symbolic cases, or run a numerical eigensolver. These roots are the eigenvalues of

T

Decode the eigenvectorsCompute a basis for each matrix eigenspace and apply

\rho_B^{-1}

to obtain eigenvectors of

T

as elements of

V

Verify in the original spaceApply

T

directly to each decoded eigenvector and confirm the result is the claimed scalar multiple.

Equations

Eigenvector equation for a transformation

EQ-EELT-01

T (\vec{v}) = λ \vec{v}, \vec{v} \in V, \vec{v} \neq \vec{0}

Stated without coordinates, so it applies to any vector space on which $T$ acts.

Eigenspace as a kernel

EQ-EELT-02

E_{T} (λ) = K (T - λ I_{V})

The eigenspace for $\lambda$ is the kernel of $T - \lambda I_V$ , hence a subspace of $V$ containing the zero vector.

Eigenvalue condition

EQ-EELT-03

λ \in spec (T) \Leftrightarrow \dim (E_{T} (λ)) \geq 1

Writing $\operatorname{spec}(T)$ for the set of eigenvalues: a scalar belongs to it exactly when its eigenspace is non-trivial, equivalently when $T - \lambda I_V$ fails to be injective. The dimension of that eigenspace is the geometric multiplicity.

Zero eigenvalue and invertibility

EQ-EELT-04

0 \in spec (T) \Leftrightarrow K (T) \neq {\vec{0}}

For a transformation of a finite-dimensional space to itself, failure of injectivity and failure of invertibility coincide.

Correspondence with the representation

EQ-EELT-05

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

Eigenvectors of the transformation correspond exactly to eigenvectors of any square representation, with the same eigenvalue.

Diagonal representation on an eigenbasis

EQ-EELT-06

B = {{\vec{v}}_{1}, \dots, {\vec{v}}_{n}} eigenvectors \Rightarrow M_{B}^{,} = diag (λ_{1}, λ_{2}, \dots, λ_{n})

If a basis of eigenvectors exists, the representation relative to it is diagonal with the eigenvalues on the diagonal.

Transposition operator

EQ-EELT-07

T (A) = A^{t} \Rightarrow A^{t} = λ A \Rightarrow A = λ^{2} A \Rightarrow λ = \pm 1

Applying the operator twice returns the original matrix, which forces the eigenvalues to satisfy $\lambda^2 = 1$ without any computation.

Variable Definitions

Symbols used on this page
Symbol	Name	Meaning	Domain / type
$T$	Linear transformation	Map from the vector space $V$ to itself	element of L(V,V)
$V$	Vector space	The space on which $T$ acts, serving as both domain and codomain	vector space of dimension n
$\vec{v}$	Eigenvector	Non-zero vector whose image under $T$ is a scalar multiple of itself	non-zero element of V
$λ$	Eigenvalue	Scale factor applied to an eigenvector by $T$	complex number
$E_{T} (λ)$	Eigenspace	All vectors satisfying the eigenvector equation for $\lambda$ , together with the zero vector	subspace of V
$I_{V}$	Identity transformation	Leaves every vector of $V$ unchanged; used to form $T - \lambda I_V$	element of L(V,V)
$M_{B}^{,}$	Square representation	Representation of $T$ using one basis $B$ at both ends	n x n matrix
$ρ_{B}$	Coordinatisation	Carries eigenvectors of $T$ to eigenvectors of the representation	invertible map V to C^n
$n$	Dimension	Dimension of $V$ , and the size of any square representation of $T$	positive integer

Worked Numerical Example

Problem statement

A stress or strain state on a planar element is stored as a $2 \times 2$ matrix, and a post-processing routine applies the transposition operator $T(A) = A^{t}$ . Find all eigenvalues and eigenvectors of this transformation directly from the definition, then build a representation on a basis of eigenvectors and interpret the result mechanically.

Confirm the setting
Let $V = M_{22}$ , the four-dimensional space of $2 \times 2$ matrices, and let $T\colon V \to V$ be $T(A) = A^{t}$ . Transposition is linear, since $(A + B)^{t} = A^{t} + B^{t}$ and $(\alpha A)^{t} = \alpha A^{t}$ , and it maps $V$ to itself, so the eigenvalue question is well posed.
Restrict the possible eigenvalues without computing
Suppose $A^{t} = \lambda A$ for some non-zero $A$ . Transposing both sides gives $A = \lambda A^{t} = \lambda^{2} A$ , and since $A \ne 0$ this forces $\lambda^{2} = 1$ . The only candidates are therefore $\lambda = 1$ and $\lambda = -1$ , established from the structure of the operator alone.
$A^{t} = λ A \Rightarrow A = λ A^{t} = λ^{2} A \Rightarrow λ^{2} = 1$
Identify the eigenspace for $\lambda = 1$
The condition $A^{t} = A$ says exactly that $A$ is symmetric. Writing a general symmetric matrix shows the eigenspace is spanned by three independent matrices, so its dimension is three.
$Identify the eigenspace for $\lambda = 1$$
Identify the eigenspace for $\lambda = -1$
The condition $A^{t} = -A$ says $A$ is skew-symmetric, which for size two forces zero diagonal entries and opposite off-diagonal entries. The eigenspace is one-dimensional.
$Identify the eigenspace for $\lambda = -1$$
Verify one eigenvector explicitly
Take $A = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$ . Transposing swaps the off-diagonal entries, giving $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ , which is exactly $(-1)A$ . The definition is satisfied with $\lambda = -1$ and the matrix is non-zero, so it is a genuine eigenvector.
Assemble an eigenbasis and its representation
The three symmetric matrices and the one skew matrix are four independent elements of a four-dimensional space, so together they form a basis $B$ of $M_{22}$ . Each is mapped to a multiple of itself, so each column of the representation has a single non-zero entry on the diagonal.
$M_{B}^{,} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - 1 \end{matrix}]$
Read the multiplicities
The eigenvalue $1$ has geometric multiplicity three and the eigenvalue $-1$ has geometric multiplicity one, summing to $4 = \dim(M_{22})$ . Because the multiplicities exhaust the dimension, an eigenbasis exists and the transformation is diagonalisable. Neither eigenvalue is zero, so the transposition operator is invertible — unsurprisingly, since it is its own inverse.
Interpret mechanically
The eigenspaces are the symmetric and skew-symmetric parts of a general $2 \times 2$ array, and the decomposition $A = \frac{1}{2}(A + A^{t}) + \frac{1}{2}(A - A^{t})$ is precisely the resolution of an arbitrary state into an eigenvector for $+1$ and an eigenvector for $-1$ . For a velocity gradient this is the split into strain rate and spin.

Result

Transposition has eigenvalues $+1$ and $-1$ with eigenspaces of dimension three and one, and relative to an eigenbasis its representation is diagonal. The classical decomposition of a tensor into symmetric and antisymmetric parts is therefore an eigenvalue decomposition, obtained here without forming a characteristic polynomial or solving a single linear system.

Applications & Industry Use

Continuum mechanics

Principal directions of stress and strain

The stress state at a point defines a linear transformation on the space of direction vectors, sending a plane normal to the traction acting on that plane. Its eigenvectors are the principal directions, on which the traction is purely normal, and its eigenvalues are the principal stresses. Failure criteria are stated in those coordinates because they are the ones intrinsic to the state rather than to the chosen axes.

Structural dynamics

Mode shapes as eigenvectors

The operator relating displacement to restoring force in a discretised structure acts on an abstract space of displacement fields. Its eigenvectors are the mode shapes and its eigenvalues determine the natural frequencies. Because eigenvalues are basis independent, a modal analysis performed in nodal coordinates and one performed in a reduced basis must agree on the frequencies.

Quantum mechanics & spectroscopy

Observables and measurement outcomes

A physical observable is represented by a linear operator on a state space, and the possible results of a measurement are its eigenvalues, with the corresponding eigenvectors as the states giving that result with certainty. The abstract formulation matters here because the state space is a space of functions, not of columns, until a basis is deliberately imposed.

Control systems

Stability from the spectrum

For a linear time-invariant system, the eigenvalues of the state transition operator determine whether trajectories grow or decay, and the eigenvectors identify the modes that do so. Because a change of state coordinates cannot change the eigenvalues, stability conclusions are safe against any rescaling or reordering of the state variables.

Data analysis

Principal components

The covariance operator of a data set acts on the space of feature vectors. Its eigenvectors are the principal directions and its eigenvalues the variances captured along them. Treating the operator abstractly clarifies why the answer is unchanged by an orthogonal change of feature basis and why it is not invariant under arbitrary rescaling of individual features.

Molecular chemistry

Symmetry operators on orbitals

Reflections and rotations of a molecule act as linear transformations on the space spanned by its atomic orbitals. Orbitals that are eigenvectors of these symmetry operators, with eigenvalues $+1$ or $-1$ , are the symmetric and antisymmetric combinations that classify bonding and antibonding states, mirroring the transposition example on this page.

Design Considerations

Try the definition before reaching for a matrix

Structured operators frequently reveal their eigenvalues by algebra alone. If applying the transformation twice returns the identity, the eigenvalues satisfy $\lambda^2 = 1$ ; if it is idempotent, they satisfy $\lambda^2 = \lambda$ ; if it is nilpotent, the only eigenvalue is zero. These arguments cost nothing, produce exact answers and often expose the eigenspaces directly.

Decide which field you are working over

The same transformation can have no eigenvalues over the real numbers and a full set over the complex numbers. A planar rotation is the standard example. Before concluding that a transformation has no eigenvalues, confirm the scalar field, and be aware that real-only software may report an empty spectrum where a complex analysis would find a conjugate pair.

Choose the basis for arithmetic convenience only

Eigenvalues do not depend on the basis used to build the representation, so the basis should be chosen to make coordinatisation and the resulting matrix as pleasant as possible. Sparse or structured representations reduce both the cost and the rounding error of the eigensolver without any risk of changing the answer.

Normalise eigenvectors before storing or comparing them

Any non-zero multiple of an eigenvector is an eigenvector, so two correct computations can return visibly different vectors. Adopt a convention — unit norm with a fixed sign, or a chosen entry set to one — and apply it consistently. Without one, regression tests on eigenvectors will fail for no mathematical reason.

Distinguish geometric from algebraic multiplicity early

An eigenvalue can be a repeated root of the characteristic polynomial while its eigenspace remains small. When the geometric multiplicities do not sum to the dimension, no eigenbasis exists and the transformation is not diagonalisable. Detecting this early avoids designing a workflow around a diagonal representation that cannot be produced.

Treat a zero eigenvalue as a structural finding

Zero in the spectrum means the transformation annihilates a non-zero vector: a mechanism in a structure, an unobservable mode in a plant, an unidentifiable parameter direction in a model. Report the corresponding eigenspace rather than merely noting that the operator is singular, since the eigenspace names exactly what is lost.

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 use of $\lambda$ for an eigenvalue, italic symbols for scalars and upright operator names, keeping the eigenvector equation typographically unambiguous.
`ISO 2041`	Mechanical vibration, shock and condition monitoring — Vocabulary	Defines the vibration terminology — natural frequency, mode shape, modal analysis — whose mathematical content is exactly the eigenvalues and eigenvectors of a structural operator.
`LAPACK / BLAS reference`	Linear Algebra PACKage reference implementation	Supplies the production eigensolvers: `xGEEV` for a general representation and `xSYEV` or `xHEEV` for symmetric and Hermitian ones, which exploit the guaranteed real spectrum.
`IEEE 754-2019`	IEEE Standard for Floating-Point Arithmetic	Determines the rounding behaviour that makes a computed eigenvalue an approximation with a backward error interpretation, and that turns exactly repeated eigenvalues into tight clusters.
`ISO/IEC 40314`	Mathematical Markup Language (MathML) Version 3.0	Encodes the eigenvector equation and the eigenspace notation as semantic markup, so the distinction between the transformation, the eigenvalue and the vector survives search indexing.

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
Complex scalars	Any transformation whose spectrum may include complex values, including rotations, oscillatory dynamics and general non-symmetric operators.	Guarantees a full complement of eigenvalues by the fundamental theorem of algebra, at the cost of complex storage and arithmetic throughout.
Real scalars only	Symmetric or self-adjoint operators, where the spectrum is provably real, and embedded contexts where complex arithmetic is unavailable.	Halves storage and simplifies interpretation, but a real-only solver will find no eigenvalues at all for genuinely rotational operators.
Exact rational arithmetic	Small symbolic operators where an exact characteristic polynomial and exact eigenspaces are required.	Produces exact multiplicities and eigenvectors, but the roots of the characteristic polynomial are generally not rational, so a symbolic or algebraic number representation is needed.
IEEE 754 binary64 with a QR-based eigensolver	Numerical spectra of moderate to large representations in engineering practice.	Backward stable and well supported, but repeated eigenvalues appear as clusters and ill-conditioned eigenvectors can be inaccurate even when the eigenvalues are not.
Symmetric or Hermitian packed storage	The representation is known to be self-adjoint with respect to the chosen inner product basis.	Roughly halves storage and enables specialised algorithms with guaranteed real eigenvalues and orthogonal eigenvectors, but the structure must genuinely hold rather than be assumed.
Interval or verified arithmetic	Safety-critical stability arguments where a rigorous bound on the spectrum is required rather than a point estimate.	Delivers a certified enclosure of each eigenvalue, at a substantial cost in speed and with the risk of pessimistic intervals for clustered spectra.

Manufacturing Notes

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

Hand procedure for a structured operator

Write the eigenvector condition as an equation among the objects of the space, not among coordinates, and solve it structurally. For transposition the condition becomes symmetry or skew-symmetry; for a projection it becomes membership of the image or of the kernel. This route produces exact eigenspaces with no characteristic polynomial and is usually faster than coordinatising.

Cost of the representation route

Building a square representation costs $n$ evaluations and $n$ coordinatisations. Reducing to Hessenberg form and running the QR algorithm costs about $O(n^3)$ for eigenvalues alone, with a further $O(n^3)$ if all eigenvectors are wanted. For $n$ up to a few thousand this is routine; beyond that, iterative methods that seek only part of the spectrum are the standard choice.

Verification in the original space

Never accept a decoded eigenvector without substituting it back into the definition. Apply $T$ to the decoded object and compare with the claimed multiple. This exercises the representation, the eigensolver and the decoding step at once, and it will catch a mismatch between the basis used for assembly and the basis used for decoding.

Library behaviour

NumPy's numpy.linalg.eig returns complex results for general matrices and normalises eigenvectors to unit length with an arbitrary phase. numpy.linalg.eigh assumes symmetry and returns sorted real eigenvalues with orthonormal eigenvectors. SymPy's eigenvects works exactly and reports algebraic and geometric multiplicities separately, which is the fastest way to detect a defective transformation.

Characteristic polynomials are for analysis, not computation

Forming the characteristic polynomial and rooting it is exact and instructive at size two or three, and numerically disastrous beyond that: the roots of a polynomial can be extremely sensitive to its coefficients even when the underlying eigenvalues are well conditioned. Use the polynomial for symbolic reasoning and multiplicity arguments, and an iterative eigensolver for numbers.

Failure Modes & Common Mistakes

Failure modes, root causes and prevention
Failure mode / mistake	Impact	Root cause	Prevention & detection
Admitting the zero vector as an eigenvector	high	Overlooking the non-zero requirement, which would make every scalar an eigenvalue.	State the requirement with the definition and check it explicitly; the eigenspace is allowed to contain the zero vector, but a reported eigenvector may not be it.
Seeking eigenvalues of a transformation between different spaces	high	Applying the definition to $T\colon U \to V$ , where $T(\vec{v})$ and $\vec{v}$ cannot be compared.	Confirm domain and codomain are the same space before starting. A rectangular representation is a clear signal that the question is ill posed.
Concluding there are no eigenvalues when working over the reals	medium	A rotation-like operator has a conjugate pair of complex eigenvalues and no real ones, and a real-only solver reports nothing.	Work over the complex field for existence arguments and interpret conjugate pairs as rotation-and-scaling in a two-dimensional invariant subspace.
Treating a zero eigenvalue as the absence of an eigenvalue	medium	Confusing the forbidden zero vector with the perfectly legitimate zero eigenvalue.	Recall that a zero eigenvalue is the signature of a non-trivial kernel, and report the kernel as its eigenspace.
Comparing eigenvectors without a normalisation convention	low	Two runs return different scalar multiples of the same direction, so an equality test fails.	Normalise to unit length with a deterministic sign or phase convention before storing or comparing.
Assuming an eigenbasis exists	medium	Designing a workflow around a diagonal representation for a transformation whose eigenspaces are too small.	Check that the geometric multiplicities sum to the dimension before relying on diagonalisation, and fall back to a triangular or block form otherwise.
Forgetting to decode the eigenvectors	high	Reporting coordinate vectors from the eigensolver as if they were elements of the abstract space.	Apply $\rho_B^{-1}$ to every eigenvector and state the answer as polynomials, matrices or whatever objects $V$ contains.
Trusting eigenvectors of a clustered spectrum	medium	Eigenvalues that are close together have eigenvectors that are extremely sensitive to perturbation, even when the eigenvalues themselves are accurate.	Report the invariant subspace spanned by the cluster instead of individual eigenvectors, and consult the eigenvalue separation before drawing directional conclusions.

FAQs

Why must an eigenvector be non-zero when the eigenvalue may be zero?

The zero vector satisfies $T(\vec{0}) = \lambda\vec{0}$ for every scalar, so allowing it would make every number an eigenvalue and the concept vacuous. A zero eigenvalue, by contrast, is a genuine statement: some non-zero vector is annihilated by the transformation, which is exactly the condition for a non-trivial kernel.

Do I need a matrix to talk about eigenvalues?

No. The definition is stated entirely in terms of the transformation and the vector space operations, and for structured operators such as transposition or projection the eigenvalues can be found from that definition alone. A matrix representation is a computational convenience for the cases where the structure is not evident.

Does the choice of basis change the eigenvalues?

No. Representations of the same transformation relative to different bases are similar matrices, and similar matrices have the same characteristic polynomial and hence the same eigenvalues. The eigenvectors change appearance, because their coordinates are expressed relative to a different basis, but they describe the same vectors in the space.

Can a linear transformation have no eigenvalues at all?

Over the complex numbers, no: any transformation of a non-zero finite-dimensional space has at least one, because the characteristic polynomial of a representation has a root. Over the real numbers, yes: a planar rotation through an angle other than a multiple of $\pi$ leaves no direction invariant. In infinite dimensions the situation is different again and requires spectral theory.

What is the relationship between eigenspaces and kernels?

An eigenspace for $\lambda$ is exactly the kernel of $T - \lambda I_V$ . This identity is worth internalising because it converts every eigenvalue computation into a kernel computation, explains why eigenspaces are subspaces, and makes the zero eigenvalue case a statement about the kernel of $T$ itself.

How do eigenvalues of a transformation relate to diagonalisation?

If enough independent eigenvectors exist to form a basis of the space, then relative to that basis the representation is diagonal with the eigenvalues on the diagonal. Diagonalisation is therefore the deliberate choice of an eigenbasis, and the failure of diagonalisation is precisely the failure of the eigenspaces to fill out the space.

Is an eigenvector unique for a given eigenvalue?

Never, since any non-zero multiple is also an eigenvector, and often not even up to scale: an eigenspace of dimension greater than one contains infinitely many independent directions, all valid. What is uniquely determined is the eigenspace itself, which is why reporting a basis for the eigenspace is more informative than reporting a single vector.

References

Beezer, R. A. A First Course in Linear Algebra, Version 0.70. University of Puget Sound, 2006. Section CB, Subsection EELT. Licensed under the GNU Free Documentation License v1.2.
Axler, S. Linear Algebra Done Right, 3rd edition. Springer, 2015.
ISO 2041:2018, Mechanical vibration, shock and condition monitoring — Vocabulary. International Organization for Standardization.
Anderson, E. et al. LAPACK Users' Guide, 3rd edition. Society for Industrial and Applied Mathematics, 1999.
Trefethen, L. N. and Bau, D. Numerical Linear Algebra. Society for Industrial and Applied Mathematics, 1997.

AI Suggested Questions

Find the eigenvalues and eigenspaces of the operator that sends a polynomial to its reflection $p(x) \mapsto p(-x)$ on the space of cubics.
Show a linear transformation of a real vector space with no real eigenvalues, and describe its invariant two-dimensional subspace.
Why does an operator satisfying $T \circ T = T$ have only the eigenvalues zero and one, and what are its eigenspaces?
Explain the connection between a zero eigenvalue, a non-trivial kernel and failure of invertibility for a transformation of a space to itself.
Compute the eigenvalues of the differentiation operator on polynomials of degree at most four and explain why the spectrum is so small.
How sensitive are eigenvectors when two eigenvalues are nearly equal, and what should be reported instead?

Related Calculators

Transformation Eigenvalue Calculator

Represent a transformation of an abstract space on a chosen basis and return its eigenvalues with decoded eigenvectors.

Eigenspace Basis Calculator

Compute a basis for the eigenspace of a given eigenvalue as the kernel of $T - \lambda I_V$ , with geometric multiplicity reported.

Characteristic Polynomial Calculator

Form the characteristic polynomial of a square representation exactly and report the algebraic multiplicity of each root.