Eigenvalues and Eigenvectors of a Matrix

Overview

Multiplying a vector $\vec{x} \in \mathbb{C}^n$ by a square matrix $A$ of size $n$ produces another vector of the same size, so the matrix can be read as a function from $\mathbb{C}^n$ to itself. In general the output points in a different direction from the input and the relationship between them is opaque; every entry of the output depends on every entry of the input. For certain exceptional inputs, however, the whole computation collapses to a single scalar multiplication: the output is a stretched, shrunk or reversed copy of the input, lying along the same line through the origin.

Those exceptional directions are the eigenvectors, and the scaling factors are the eigenvalues. The two always come as a pair, because the equation $A\vec{x} = \lambda\vec{x}$ constrains them jointly. The definition insists that $\vec{x} \ne \vec{0}$ , since the zero vector satisfies the equation for every conceivable $\lambda$ and would render the notion empty. It places no such restriction on $\lambda$ : an eigenvalue of zero is perfectly admissible and is exactly the signature of a singular matrix.

Eigenpairs are not a curiosity. They are the reason a coupled system of differential equations can be decoupled, the reason a structure has natural frequencies, the reason a Markov chain has a steady state and the reason a covariance matrix has principal directions. In each case a complicated interaction is re-expressed in a basis of directions in which the matrix acts by pure scaling, and the analysis becomes one-dimensional in each direction independently.

Two structural facts emerge immediately from the definition, before any computational machinery arrives. First, eigenvectors are never unique: any non-zero scalar multiple of an eigenvector is an eigenvector with the same eigenvalue. Second, the eigenvectors belonging to one fixed eigenvalue, together with the zero vector, are closed under addition and scalar multiplication — they form a subspace. Both observations follow directly from the linearity of the matrix-vector product, and both shape how eigenvectors are reported and normalised in practice.

Definition

Eigenvalues and Eigenvectors of a Matrix

EEM

Let $A$ be a square matrix of size $n$ , let $\vec{x} \in \mathbb{C}^n$ with $\vec{x} \ne \vec{0}$ , and let $\lambda \in \mathbb{C}$ . Then $\vec{x}$ is an eigenvector of $A$ with eigenvalue $\lambda$ when

$A\vec{x} = \lambda\vec{x}$ .

The requirement

\vec{x} \ne \vec{0}

is part of the definition. No restriction is placed on

\lambda

, which may be zero, negative, irrational or complex.

The condition is a statement about the pair, not about either object alone. It is meaningless to call a vector an eigenvector without naming the matrix, and meaningless to call a scalar an eigenvalue without exhibiting some non-zero vector that realises it.

Eigenpair

The ordered pair $(\lambda, \vec{x})$ satisfying $A\vec{x} = \lambda\vec{x}$ with $\vec{x} \ne \vec{0}$ . Naming the pair emphasises that the eigenvalue and eigenvector are determined together; software routines almost always return them together for this reason.

Spectrum of a Matrix

SPEC

The set of all eigenvalues of $A$ , written $\sigma(A)$ . For a matrix of size $n$ the spectrum is a non-empty set containing at most $n$ distinct complex numbers. The spectral radius is $\max\{\,|\lambda| : \lambda \in \sigma(A)\,\}$ , the largest eigenvalue magnitude, which governs the growth or decay of $A^{k}$ as $k$ increases.

Concepts

The functional reading of a matrix

Write $T(\vec{x}) = A\vec{x}$ . This is a function from $\mathbb{C}^n$ to $\mathbb{C}^n$ , and it is linear: $T(\alpha\vec{u} + \beta\vec{v}) = \alpha T(\vec{u}) + \beta T(\vec{v})$ . The eigenvector question asks which inputs the function leaves on their own line through the origin. Such a direction is invariant under $T$ : the one-dimensional subspace $\operatorname{span}(\{\vec{x}\})$ is mapped into itself, and on that subspace the whole matrix acts as multiplication by the single number $\lambda$ .

Why the eigenvector must be non-zero and the eigenvalue need not be

If $\vec{x} = \vec{0}$ were permitted, then $A\vec{0} = \vec{0} = \lambda\vec{0}$ holds for every scalar $\lambda$ whatsoever, so every complex number would be an eigenvalue of every matrix and the concept would carry no information. Excluding the zero vector is therefore not a technicality but the substance of the definition. The eigenvalue, by contrast, is unconstrained: $\lambda = 0$ means $A\vec{x} = \vec{0}$ for some non-zero $\vec{x}$ , which is precisely the statement that $A$ is singular. Zero is an eigenvalue of $A$ if and only if $A$ is singular.

Eigenvectors come in families

Suppose $A\vec{x} = \lambda\vec{x}$ and let $\alpha \ne 0$ . Then $A(\alpha\vec{x}) = \alpha(A\vec{x}) = \alpha\lambda\vec{x} = \lambda(\alpha\vec{x})$ , so $\alpha\vec{x}$ is an eigenvector for the same eigenvalue. An eigenvector is therefore never unique, only its direction is determined, and any reported eigenvector is one representative of an infinite family. Numerical libraries resolve the ambiguity by normalising to unit length, which still leaves a sign (or, over $\mathbb{C}$ , a phase) undetermined — a fact that surprises anyone comparing eigenvector output across two implementations.

The eigenvectors for one eigenvalue form a subspace

If $A\vec{u} = \lambda\vec{u}$ and $A\vec{v} = \lambda\vec{v}$ for the same $\lambda$ , then $A(\vec{u} + \vec{v}) = A\vec{u} + A\vec{v} = \lambda\vec{u} + \lambda\vec{v} = \lambda(\vec{u} + \vec{v})$ . The sum is an eigenvector for $\lambda$ unless it happens to be zero. Adjoining the zero vector to the set of eigenvectors for $\lambda$ therefore produces a set closed under addition and scalar multiplication — a subspace of $\mathbb{C}^n$ . Its dimension can exceed one, which is the situation in which two eigenvectors share an eigenvalue without being multiples of one another. Note that this closure fails across different eigenvalues: adding eigenvectors for $\lambda_1 \ne \lambda_2$ produces a vector that is generally not an eigenvector at all.

Rearranging into a null space question

The defining equation can be rewritten. From $A\vec{x} = \lambda\vec{x}$ , subtract the right-hand side and insert the identity: $A\vec{x} - \lambda I_n\vec{x} = \vec{0}$ , hence $(A - \lambda I_n)\vec{x} = \vec{0}$ . So $\vec{x}$ is a non-zero element of the null space $\mathcal{N}(A - \lambda I_n)$ . This is the pivot on which all computation turns: $\lambda$ is an eigenvalue exactly when $A - \lambda I_n$ is singular, and once $\lambda$ is known the eigenvectors are found by solving a homogeneous system. The subspace of the previous concept is now identified concretely as $\mathcal{N}(A - \lambda I_n)$ .

Why the scalars must be complex

Restricting attention to real scalars would leave many perfectly ordinary matrices with no eigenvalues at all. The matrix representing a quarter-turn rotation of the plane, $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ , maps no real direction onto itself — geometrically obvious, since every direction is rotated by ninety degrees. Over $\mathbb{C}$ it has the eigenvalues $i$ and $-i$ . Working with complex scalars from the outset guarantees that every square matrix has at least one eigenvalue, and complex eigenvalues of real matrices carry genuine engineering meaning: they encode oscillation, with the imaginary part giving frequency and the real part giving growth or decay.

Testing whether a candidate is an eigenvector

Check the vector is non-zeroThe zero vector is excluded by definition. A candidate that is

\vec{0}

fails immediately, whatever the arithmetic says.

Form the product $A\vec{x}$ One matrix-vector multiplication, costing about

2n^2

operations. No system is solved.

Is the output a scalar multiple of the input?Compare entry by entry. If any two entries give inconsistent ratios, the candidate is not an eigenvector.

Read off the eigenvalueThe common ratio is

\lambda

. Use an entry of

\vec{x}

that is non-zero to compute it, and confirm the same value from a second entry.

Interpret $\lambda$

\lambda = 0

means

\vec{x}

lies in the null space and

A

is singular.

|\lambda| > 1

means the direction is amplified by repeated application;

|\lambda| < 1

means it decays.

Equations

Defining relation for an eigenpair

EQ-EEM-01

A \vec{x} = λ \vec{x}, \vec{x} \neq \vec{0}

The matrix-vector product on the left collapses to a scalar multiplication on the right. The non-zero condition on $\vec{x}$ is part of the definition, not an afterthought.

Homogeneous form

EQ-EEM-02

(A - λ I_{n}) \vec{x} = \vec{0}, \vec{x} \neq \vec{0}

Obtained by subtracting $\lambda I_n \vec{x}$ from both sides. This form converts the eigenvector question into a null space computation.

Eigenvectors as a null space

EQ-EEM-03

\vec{x} \in N (A - λ I_{n}), λ \in σ (A) \Leftrightarrow A - λ I_{n} is singular

A scalar is an eigenvalue exactly when the shifted matrix has a non-trivial null space. Adjoining $\vec{0}$ to the eigenvectors for $\lambda$ gives precisely this null space.

Closure of an eigenvector family

EQ-EEM-04

A \vec{u} = λ \vec{u}, A \vec{v} = λ \vec{v} \Rightarrow A (α \vec{u} + β \vec{v}) = λ (α \vec{u} + β \vec{v})

Any linear combination of eigenvectors sharing an eigenvalue is again an eigenvector for that eigenvalue, provided the combination is not the zero vector.

Zero eigenvalue and singularity

EQ-EEM-05

0 \in σ (A) \Leftrightarrow N (A) \neq {\vec{0}} \Leftrightarrow A is singular \Leftrightarrow \det (A) = 0

The chain that connects the eigenvalue zero to every other characterisation of singularity. The null space of $A$ is exactly the eigenspace for $\lambda = 0$ , plus the zero vector.

Repeated application

EQ-EEM-06

A \vec{x} = λ \vec{x} \Rightarrow A^{k} \vec{x} = λ^{k} \vec{x}

Along an eigendirection, applying the matrix $k$ times multiplies by $\lambda^{k}$ . This is why the spectral radius governs the long-run behaviour of iterations, power methods and discrete-time state models.

A real matrix with no real eigenvalue

EQ-EEM-07

[\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}] [\begin{matrix} 1 \\ - i \end{matrix}] = [\begin{matrix} i \\ 1 \end{matrix}] = i [\begin{matrix} 1 \\ - i \end{matrix}]

The quarter-turn rotation of the plane. It maps no real direction to a multiple of itself, but over $\mathbb{C}$ it has the eigenpair shown, together with its conjugate for $\lambda = -i$ .

Variable Definitions

Symbols used on this page
Symbol	Name	Meaning	Domain / type
$A$	Matrix	The square matrix whose eigenpairs are sought	n x n matrix over C
$n$	Size	Common row and column count of $A$ ; also the number of entries in each eigenvector	positive integer
$\vec{x}$	Eigenvector	A non-zero vector whose image under $A$ is a scalar multiple of itself	non-zero vector in C^n
$λ$	Eigenvalue	The scaling factor applied by $A$ along the direction $\vec{x}$	complex scalar, possibly zero
$I_{n}$	Identity matrix	Used to form the shifted matrix $A - \lambda I_n$	n x n matrix
$σ (A)$	Spectrum	The set of all eigenvalues of $A$	non-empty subset of C, at most n elements
$N (A - λ I_{n})$	Eigenspace	The eigenvectors for $\lambda$ together with the zero vector	subspace of C^n
$α$	Scalar multiplier	Any non-zero scalar used to rescale an eigenvector	non-zero complex scalar

Worked Numerical Example

Problem statement

A triangular network of three nodes is connected pairwise by identical conductances of $2\ \mathrm{S}$ . The nodal admittance matrix relates the vector of node potentials to the vector of injected currents. Identify eigenpairs of this matrix by direct testing, and interpret what each one says about the network.

Assemble the admittance matrix
Each node connects to the other two through a conductance of $2\ \mathrm{S}$ , so each diagonal entry is the sum of the conductances at that node and each off-diagonal entry is the negative of the connecting conductance. The relation is $Y\vec{v} = \vec{i}$ , with $\vec{v}$ the node potentials and $\vec{i}$ the injected currents.
$Y = [\begin{matrix} 4 & - 2 & - 2 \\ - 2 & 4 & - 2 \\ - 2 & - 2 & 4 \end{matrix}]$
Test a uniform potential vector
Take $\vec{u}$ with all three nodes at the same potential. Each row of the product sums to $4 - 2 - 2 = 0$ , so the output is the zero vector, which equals $0 \cdot \vec{u}$ .
$Y [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}] = 0 \cdot [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]$
Record the first eigenpair and its consequence
The vector is non-zero, so it qualifies as an eigenvector, with eigenvalue $\lambda_1 = 0$ . Because zero is an eigenvalue, $Y$ is singular: no unique potential solution exists for a given current injection until one node is chosen as a reference. Physically, raising every node potential by the same amount drives no current, which is exactly what the eigenvector says.
Test a differential pattern
Take a vector representing two nodes at opposite potentials with the third at zero. The three output entries are $4 + 2 + 0 = 6$ , $-2 - 4 + 0 = -6$ and $-2 + 2 + 0 = 0$ , so the output is six times the input.
$Y [\begin{matrix} 1 \\ - 1 \\ 0 \end{matrix}] = [\begin{matrix} 6 \\ - 6 \\ 0 \end{matrix}] = 6 [\begin{matrix} 1 \\ - 1 \\ 0 \end{matrix}]$
Find a second, independent eigenvector for the same eigenvalue
A different differential pattern gives $0 - 4 - 2 \cdot(-1) = -2 + 2 = 0$ in the first entry, $0 + 4 + 2 = 6$ in the second, and $0 - 2 - 4 = -6$ in the third. This vector is not a scalar multiple of the previous one, yet it has the same eigenvalue $6$ .
$Y [\begin{matrix} 0 \\ 1 \\ - 1 \end{matrix}] = [\begin{matrix} 0 \\ 6 \\ - 6 \end{matrix}] = 6 [\begin{matrix} 0 \\ 1 \\ - 1 \end{matrix}]$
Confirm closure under addition
Adding the two eigenvectors for $\lambda = 6$ must give another eigenvector for $\lambda = 6$ , by linearity of the matrix-vector product. Direct computation confirms it: the entries of the product are $4 + 0 + 2 = 6$ , $-2 + 0 + 2 = 0$ and $-2 + 0 - 4 = -6$ .
$Y [\begin{matrix} 1 \\ 0 \\ - 1 \end{matrix}] = [\begin{matrix} 6 \\ 0 \\ - 6 \end{matrix}] = 6 [\begin{matrix} 1 \\ 0 \\ - 1 \end{matrix}]$
Confirm closure under scaling
Scaling the first eigenvector by $5$ gives another eigenvector for $\lambda = 0$ , since $Y(5\vec{u}) = 5(Y\vec{u}) = 5 \cdot \vec{0} = \vec{0}$ . Every non-zero multiple works, so the eigenvector is determined only up to scale.
Test a vector that is not an eigenvector
Not every direction is special. Injecting a unit potential at node one alone gives an output whose entries are in the ratios $4 : -2 : -2$ against an input of $1 : 0 : 0$ . No single scalar reproduces that, so this vector is not an eigenvector of $Y$ .
$Y [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}] = [\begin{matrix} 4 \\ - 2 \\ - 2 \end{matrix}] \neq λ [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}] for any λ$

Result

The network has two distinct eigenvalues: $\lambda_1 = 0$ with the single independent eigendirection of uniform potential, and $\lambda_2 = 6$ with a two-dimensional family of differential patterns. Their sum, $0 + 6 + 6 = 12$ , matches the trace of $Y$ , an independent arithmetic check. The engineering reading is direct: the zero eigenvalue is the floating-reference degeneracy that every nodal analysis must resolve by grounding a node, while the repeated eigenvalue $6$ reflects the symmetry of the triangle — every differential excitation, regardless of which pair of nodes it drives, sees the same effective admittance of $6\ \mathrm{S}$ .

Applications & Industry Use

Structural dynamics

Natural frequencies and mode shapes

For an undamped structure, $K\vec{\phi} = \omega^{2}M\vec{\phi}$ is a generalised eigenvalue problem whose eigenvalues are the squared natural frequencies and whose eigenvectors are the mode shapes. A structure excited exactly along a mode shape responds in that shape alone, which is the whole basis of modal testing and of resonance avoidance in design.

Control systems

Stability of a state-space model

The state matrix $A$ of a linear model determines stability entirely through its eigenvalues: a continuous-time system is asymptotically stable when every eigenvalue has negative real part, and a discrete-time system when every eigenvalue has modulus less than one. Complex eigenvalues correspond to oscillatory modes, with the imaginary part setting the frequency and the real part the envelope.

Electrical power systems

Network Laplacians and reference nodes

A nodal admittance matrix built without a ground reference always has zero as an eigenvalue, with the uniform-potential vector as its eigenvector. Recognising this eigenpair explains why a load-flow solver must fix a slack bus, and the multiplicity of the zero eigenvalue counts the number of electrically disconnected islands in the network.

Data science and metrology

Principal directions of a covariance matrix

The eigenvectors of a covariance matrix are the uncorrelated directions of a data set and the eigenvalues are the variances along them. A near-zero eigenvalue identifies a direction of no variation, which in instrumentation practice signals a redundant sensor channel or a constraint the measurement system has not been told about.

Chemical engineering

Reaction network time scales

Linearising a reaction network about an operating point gives a Jacobian whose eigenvalues are reciprocal time constants. Widely separated eigenvalues mean a stiff system in which fast species equilibrate almost instantly, which both dictates the choice of integrator and justifies quasi-steady-state reduction of the fast modes.

Computer graphics and mechanics

Principal axes of inertia and strain

The inertia tensor and the strain tensor are symmetric matrices whose eigenvectors are the principal axes and whose eigenvalues are the principal moments or principal strains. Working in the principal frame diagonalises the tensor, so rotational dynamics and failure criteria both simplify to independent one-dimensional statements.

Design Considerations

Report eigenvectors with an explicit normalisation

Because every non-zero multiple of an eigenvector is an eigenvector, a bare eigenvector is ambiguous. Fix a convention and state it: unit Euclidean norm, largest entry equal to one, or first non-zero entry positive. Without one, two correct implementations will produce outputs that appear to disagree, and any downstream comparison or regression test becomes unreliable.

Expect a sign or phase ambiguity even after normalising

Unit norm still leaves $\pm\vec{x}$ over the reals, and any unit-modulus multiple over $\mathbb{C}$ . Comparisons between eigenvector sets must therefore be made up to sign, typically by comparing the absolute value of the inner product or by using a subspace-distance measure rather than an entrywise difference.

Never test $\lambda = 0$ against exact zero in floating point

A singular matrix computed in floating point returns an eigenvalue of small magnitude rather than exactly zero. Judge it against a scaled tolerance, typically a multiple of the unit roundoff times the matrix norm, and prefer the smallest singular value if the real question is distance to singularity.

Distinguish repeated eigenvalues from close ones

Two eigenvectors sharing an eigenvalue span a plane in which every non-zero vector is an eigenvector, so the individual eigenvectors returned by a solver are arbitrary within that plane. Two eigenvalues that are merely close have well-defined but extremely sensitive eigenvectors. The two situations look similar numerically and demand different handling — work with the invariant subspace rather than with individual eigenvectors.

Keep the eigenvalue problem in complex arithmetic

A real matrix can have complex eigenvalues, always in conjugate pairs. Truncating to real output silently discards oscillatory modes, which are usually the ones that matter for stability and vibration. If a real-arithmetic representation is required, use the real Schur form with $2 \times 2$ blocks rather than discarding the imaginary parts.

Exploit structure before calling a general solver

Symmetric and Hermitian matrices have real eigenvalues and an orthonormal eigenvector basis, and their specialised solvers are faster, more accurate and better behaved under perturbation than the general non-symmetric routines. Enforcing symmetry explicitly — replacing $A$ by $\tfrac{1}{2}(A + A^{t})$ when it should be symmetric — is usually worth the cost.

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	Establishes $\lambda$ as the conventional symbol for an eigenvalue and governs the italic and upright typography distinguishing the scalar $\lambda$ from operator names.
`LAPACK xGEEV / xSYEV`	Eigenvalue and eigenvector driver routines	The reference interfaces for the general and symmetric eigenproblems. Their documented normalisation — unit Euclidean norm with the largest component real — is the de facto convention for reported eigenvectors.
`IEEE 754-2019`	IEEE Standard for Floating-Point Arithmetic	Defines the arithmetic in which eigenvalues are computed and hence why a structurally zero eigenvalue appears as a small non-zero number requiring a tolerance.
`ISO 18431-1`	Mechanical vibration and shock — Signal processing	Frames modal analysis practice, in which measured natural frequencies and mode shapes are the eigenvalues and eigenvectors of the structural model being validated.
`IEC 61970 (CIM)`	Energy management system application program interface	Underlies the network models on which power-system eigenvalue analysis is performed, including the connectivity information that determines the multiplicity of the zero eigenvalue of a network Laplacian.

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	Small symbolic matrices where eigenvalues are wanted as exact algebraic numbers or as expressions in a parameter.	Exact and unambiguous, but eigenvalues are roots of a polynomial and are generally irrational even for rational input, so the exactness ends at the characteristic polynomial.
IEEE 754 binary64, general (non-symmetric) solver	Arbitrary square matrices from state-space models, Jacobians and transition matrices.	Handles every case and returns complex eigenpairs, but is slower than the symmetric path and the eigenvectors can be very ill conditioned when eigenvalues are close.
IEEE 754 binary64, symmetric or Hermitian solver	Covariance, stiffness, inertia and Laplacian matrices that are symmetric by construction.	Faster, more accurate, real eigenvalues and an orthonormal eigenvector basis, but silently gives wrong answers if the input is not actually symmetric.
IEEE 754 binary32	Large-scale or accelerator-based computations where only the dominant eigenvalues are needed to modest accuracy.	Halves memory traffic, but roughly seven significant digits leaves little margin for the eigenvalue sensitivity of a non-normal matrix.
Sparse iterative methods (Arnoldi, Lanczos)	Very large sparse matrices where only a few eigenvalues at one end of the spectrum are required.	Never forms a dense factorisation and scales to millions of unknowns, but returns only part of the spectrum and requires shift-and-invert to target interior eigenvalues.
Interval or verified arithmetic	Safety-critical stability certification where a rigorous enclosure of the eigenvalues is required rather than an estimate.	Delivers a mathematically guaranteed bound, at a cost of one to two orders of magnitude in runtime and considerable implementation complexity.

Manufacturing Notes

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

Cost of testing versus finding

Verifying a proposed eigenpair costs one matrix-vector product, about $2n^{2}$ operations. Finding the eigenpairs from scratch costs an order of magnitude more: reduction to Hessenberg form followed by the shifted QR algorithm runs at roughly $10n^{3}$ operations for eigenvalues alone and about $25n^{3}$ when eigenvectors are also required. Where a candidate eigenvector is available from physical reasoning, testing it is far cheaper than solving the general problem.

Hand procedure for verification

Compute $A\vec{x}$ and divide entrywise by $\vec{x}$ , skipping any zero entries of $\vec{x}$ . All surviving ratios must agree; that common value is $\lambda$ . If any two disagree, the candidate is not an eigenvector, and no amount of rescaling will make it one. Checking a second entry independently catches the most common arithmetic slip.

Library behaviour and normalisation

numpy.linalg.eig returns eigenvalues in no guaranteed order with unit-norm eigenvectors as the columns of the returned array — a frequent source of transposition errors. numpy.linalg.eigh is the symmetric or Hermitian path and returns real eigenvalues in ascending order. MATLAB's eig and SciPy's scipy.linalg.eig follow the same LAPACK conventions. SymPy's Matrix.eigenvects() returns exact eigenvalues with their algebraic multiplicities and basis vectors for each eigenspace.

Verification of a computed eigenpair

Compute the residual $\|A\vec{x} - \lambda\vec{x}\|$ and compare it against $\|A\|\,\|\vec{x}\|$ times the unit roundoff. A small residual certifies that the pair is an exact eigenpair of a nearby matrix, which is the strongest guarantee available in floating point. Note that a small residual does not imply the eigenvector is close to a true eigenvector when the eigenvalue is poorly separated.

Sanity checks from trace and determinant

The sum of the eigenvalues, with algebraic multiplicity, equals the trace of the matrix; their product equals the determinant. Both are cheap to compute independently and catch gross errors in an eigenvalue computation immediately. For the triangular network above, $0 + 6 + 6 = 12$ matches the trace and $0 \times 6 \times 6 = 0$ matches the determinant of the singular matrix.

Failure Modes & Common Mistakes

Failure modes, root causes and prevention
Failure mode / mistake	Impact	Root cause	Prevention & detection
Accepting the zero vector as an eigenvector	high	Solving $(A - \lambda I_n)\vec{x} = \vec{0}$ and reporting the trivial solution, which satisfies the equation for every $\lambda$ .	Require a non-trivial null space. If the only solution is $\vec{x} = \vec{0}$ then $\lambda$ is not an eigenvalue at all.
Assuming $\lambda = 0$ is not allowed	medium	Reading the non-zero restriction as applying to the eigenvalue rather than the eigenvector.	Remember that $\lambda = 0$ is the signature of a singular matrix and that its eigenspace is exactly the null space of $A$ .
Comparing eigenvectors entrywise across implementations	medium	Ignoring that eigenvectors are determined only up to a non-zero scalar, and up to sign or phase even after normalisation.	Compare directions, not entries: use $\|\langle \vec{x}_1, \vec{x}_2 \rangle\|$ against the product of the norms, or compare the invariant subspaces.
Adding eigenvectors belonging to different eigenvalues	medium	Over-generalising the closure property, which holds only within a single eigenvalue.	Verify the eigenvalues agree before combining. A sum across distinct eigenvalues is almost never an eigenvector.
Discarding complex eigenvalues of a real matrix	high	Casting solver output to a real type, which silently deletes the oscillatory modes that usually dominate the dynamic behaviour.	Keep complex arithmetic throughout, or use the real Schur form with $2 \times 2$ blocks that represent conjugate pairs.
Using a symmetric solver on a non-symmetric matrix	high	Calling a routine such as `eigh` on a matrix that is only approximately symmetric; the routine reads one triangle and ignores the other.	Test symmetry explicitly against a tolerance, and either symmetrise deliberately or dispatch to the general solver.
Treating a repeated eigenvalue's eigenvectors as canonical	medium	Relying on the specific basis a solver returns for a multi-dimensional eigenspace, which is arbitrary within that space.	Work with the eigenspace as a subspace. Any quantity that depends on which basis was returned is not a well-defined property of the matrix.
Applying the definition to a non-square matrix	low	Attempting an eigenvalue analysis on a rectangular matrix, where $A\vec{x}$ and $\vec{x}$ do not even have the same size.	Eigenvalues require a square matrix. For rectangular matrices the analogous objects are the singular values and singular vectors.

FAQs

Why must an eigenvector be non-zero?

Because $A\vec{0} = \vec{0} = \lambda\vec{0}$ holds for every scalar $\lambda$ . If the zero vector were admitted, every complex number would be an eigenvalue of every matrix and the definition would convey nothing. Excluding it is what makes the existence of an eigenpair a genuine statement about the matrix.

Can an eigenvalue be zero?

Yes, and it is an important case. An eigenvalue of zero means there is a non-zero $\vec{x}$ with $A\vec{x} = \vec{0}$ , so the null space is non-trivial and the matrix is singular. Equivalently, zero is an eigenvalue exactly when $\det(A) = 0$ . The eigenspace for $\lambda = 0$ is precisely the null space of $A$ .

How many eigenvectors does one eigenvalue have?

Infinitely many, since every non-zero multiple of an eigenvector is another eigenvector for the same eigenvalue. The meaningful question is how many linearly independent ones there are, which is the dimension of the eigenspace $\mathcal{N}(A - \lambda I_n)$ . That dimension is at least one and at most the multiplicity of $\lambda$ as a root of the characteristic polynomial.

Can two different eigenvectors share an eigenvalue without being multiples of each other?

Yes. When the eigenspace for $\lambda$ has dimension two or more, it contains independent eigenvectors, and every non-zero vector in that subspace is an eigenvector for $\lambda$ . This is what a symmetric structure typically produces — the triangular network in the worked example has a two-dimensional eigenspace for its non-zero eigenvalue because of its threefold symmetry.

Why are the scalars taken from the complex numbers?

Because over the reals a perfectly ordinary matrix can have no eigenvalues at all — a planar rotation is the standard example, since it leaves no real direction invariant. Over $\mathbb{C}$ every square matrix has at least one eigenvalue. Complex eigenvalues of real matrices are also physically meaningful: they arrive in conjugate pairs and encode oscillation.

Does every square matrix have a full set of independent eigenvectors?

No. Every square matrix has at least one eigenvalue and hence at least one eigenvector, but the eigenvectors need not span the whole space. A matrix such as $\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ has the single eigenvalue $1$ with only a one-dimensional eigenspace. Matrices whose eigenvectors do span are called diagonalisable, and they are the ones for which the eigen-decomposition is a complete description.

What is the practical difference between testing an eigenpair and computing one?

Testing is one matrix-vector product, roughly $2n^{2}$ operations, and needs no solver at all. Computing the eigenpairs of a general matrix requires reduction to Hessenberg form and an iterative QR process, costing on the order of $10n^{3}$ to $25n^{3}$ operations. When physical reasoning suggests a candidate — a uniform potential, a rigid-body mode, a symmetry direction — testing it directly is far cheaper than a full decomposition.

References

Beezer, R. A. A First Course in Linear Algebra, Version 0.70. University of Puget Sound, 2006. Section EE, Subsection EEM. Licensed under the GNU Free Documentation License v1.2.
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.
Golub, G. H. and Van Loan, C. F. Matrix Computations, 4th edition. Johns Hopkins University Press, 2013. Chapters 7 and 8.
Horn, R. A. and Johnson, C. R. Matrix Analysis, 2nd edition. Cambridge University Press, 2013.

AI Suggested Questions

Give me a $3 \times 3$ real matrix with one real eigenvalue and a complex conjugate pair, and explain what the complex pair means for the dynamics of the continuous-time system with state matrix $A$ .
Show a matrix whose eigenspace for a repeated eigenvalue has dimension one rather than two, and explain why the eigenvectors fail to span.
Why does the multiplicity of the zero eigenvalue of a network Laplacian equal the number of connected components?
Compare the cost of verifying a candidate eigenpair against computing the full spectrum for a matrix of size 1000.
How sensitive are eigenvectors to perturbations of the matrix when two eigenvalues are close, and what quantity bounds that sensitivity?
Explain the relationship between the eigenvalues of $A$ and the singular values of $A$ , and why they coincide for symmetric positive definite matrices.

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Compute the full spectrum of a square matrix with eigenvectors, multiplicities and residual checks.

Eigenpair Verifier

Test whether a proposed vector is an eigenvector of a given matrix and report the eigenvalue and the residual.

Characteristic Polynomial Calculator

Expand $\det(A - xI_n)$ symbolically and factor it to expose the eigenvalues and their algebraic multiplicities.

Overview

Definition

Eigenvalues and Eigenvectors of a Matrix

Eigenpair

Spectrum of a Matrix

Concepts

The functional reading of a matrix

Why the eigenvector must be non-zero and the eigenvalue need not be

Eigenvectors come in families

The eigenvectors for one eigenvalue form a subspace

Rearranging into a null space question

Why the scalars must be complex

Testing whether a candidate is an eigenvector

Equations

Variable Definitions

Worked Numerical Example

Problem statement

Assemble the admittance matrix

Test a uniform potential vector

Record the first eigenpair and its consequence

Test a differential pattern

Find a second, independent eigenvector for the same eigenvalue

Confirm closure under addition

Confirm closure under scaling

Test a vector that is not an eigenvector

Applications &amp; Industry Use

Natural frequencies and mode shapes

Stability of a state-space model

Network Laplacians and reference nodes

Principal directions of a covariance matrix

Reaction network time scales

Principal axes of inertia and strain

Design Considerations

Report eigenvectors with an explicit normalisation

Expect a sign or phase ambiguity even after normalising

Never test λ=0 against exact zero in floating point

Distinguish repeated eigenvalues from close ones

Keep the eigenvalue problem in complex arithmetic

Exploit structure before calling a general solver

Standards &amp; Codes

Material Selection

Manufacturing Notes

Cost of testing versus finding

Hand procedure for verification

Library behaviour and normalisation

Verification of a computed eigenpair

Sanity checks from trace and determinant

Failure Modes &amp; Common Mistakes

FAQs

Related Engineering Topics

References

AI Suggested Questions

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Applications & Industry Use

Never test $\lambda = 0$ against exact zero in floating point

Standards & Codes

Failure Modes & Common Mistakes