Computing Eigenvalues and Eigenvectors

Q: Why must the identity matrix be inserted when moving from $A\vec{x} = \lambda\vec{x}$ to $(A - \lambda I_n)\vec{x} = \vec{0}$?

The expression $A - \lambda$ is not defined: a matrix and a scalar cannot be subtracted. Writing $\lambda\vec{x}$ as $\lambda I_n \vec{x}$ turns the right-hand side into a matrix acting on $\vec{x}$, and only then does the distributive law for matrix-vector products allow $\vec{x}$ to be factored out. Skipping this step produces the correct-looking answer for the wrong reason and leads directly to the error of subtracting $\lambda$ from every entry.

Q: Can a matrix with all real entries have complex eigenvalues?

Yes, and it is common. A plane rotation through an angle that is not a multiple of $\pi$ maps no real direction to a multiple of itself, so it has no real eigenvalue at all; its characteristic polynomial is an irreducible real quadratic with a conjugate pair of complex roots. For a real matrix, complex eigenvalues always occur in conjugate pairs, and the corresponding eigenvectors are conjugates of one another.

Q: Is the eigenspace for $\lambda$ the same as the set of eigenvectors for $\lambda$?

Almost. The eigenspace is the set of eigenvectors together with the zero vector. That single extra element is what makes the set closed under scalar multiplication and hence a subspace, which in turn gives it a dimension and a basis. The zero vector itself is never an eigenvector.

Q: How many eigenvalues can an $n \times n$ matrix have?

At most $n$ distinct ones. The characteristic polynomial has degree exactly $n$, so counted with algebraic multiplicity there are exactly $n$ roots in $\mathbb{C}$, and distinct roots therefore number at most $n$. There is always at least one, because a non-constant polynomial always has a complex root.

Q: Why do numerical libraries avoid the characteristic polynomial?

Because the map from matrix entries to polynomial coefficients, and from coefficients to roots, is severely ill-conditioned. A relative perturbation of size $\varepsilon$ in a coefficient can shift a root of multiplicity $m$ by roughly $\varepsilon^{1/m}$, so even a perfectly accurate root-finder returns poor eigenvalues. The QR iteration works directly on the matrix and is backward stable, which is why every production library uses it.

Q: What does it mean if $\lambda = 0$ turns out to be an eigenvalue?

It means $A - 0 I_n = A$ is singular, so the matrix has a non-trivial null space and is not invertible. The eigenspace for $\lambda = 0$ is exactly $\mathcal{N}(A)$. Equivalently, $\det(A) = 0$, which is visible as a zero constant term in the characteristic polynomial.

Q: Do I need to normalise the eigenvectors I compute?

Not for correctness — any non-zero multiple of an eigenvector is an eigenvector for the same eigenvalue. Normalisation matters for presentation and for comparison. By hand, clear fractions and present the smallest integer representative; numerically, scale to unit norm and fix a sign or phase convention so results are reproducible across runs and libraries.

Overview

The defining equation of an eigenvalue, $A\vec{x} = \lambda\vec{x}$ with $\vec{x} \ne \vec{0}$ , looks unusable as a computational recipe: two unknowns appear multiplied together, and the scalar $\lambda$ is buried inside a vector equation. The first move is purely algebraic and it dissolves the difficulty entirely. Subtract, insert the identity, and factor — the equation becomes $(A - \lambda I_n)\vec{x} = \vec{0}$ , a homogeneous linear system whose coefficient matrix depends on $\lambda$ .

That single rearrangement splits the problem into two independent stages. Stage one asks for which values of $\lambda$ the homogeneous system has a non-trivial solution — equivalently, for which $\lambda$ the matrix $A - \lambda I_n$ is singular. Because a square matrix is singular exactly when its determinant vanishes, stage one is a scalar root-finding problem: solve $\det(A - \lambda I_n) = 0$ . Stage two, for each root found, asks for the actual non-zero solutions, which is a null space computation and therefore ordinary row reduction.

The determinant $\det(A - xI_n)$ , treated as a function of the indeterminate $x$ , is the characteristic polynomial $p_A(x)$ . For an $n \times n$ matrix it has degree exactly $n$ , so the fundamental theorem of algebra guarantees $n$ roots counted with multiplicity in $\mathbb{C}$ . This is why the theory of eigenvalues is stated over the complex numbers even when the matrix has real entries: a real matrix can easily have no real eigenvalue at all, but it can never have no complex eigenvalue.

The eigenvectors for a fixed eigenvalue $\lambda$ , together with the zero vector, form the eigenspace $\mathcal{E}_A(\lambda)$ . Identifying it as $\mathcal{N}(A - \lambda I_n)$ is more than a change of name: it certifies that the eigenspace is a subspace, gives it a dimension, and hands over the entire null space toolkit — reduce, read the free variables, write a basis. Every eigenvector for $\lambda$ is then a linear combination of a handful of basis vectors, and the infinite set of eigenvectors is described by finitely many numbers.

Definition

Eigenvalue and Eigenvector of a Matrix

EEM

Let $A$ be a square matrix of size $n$ . A scalar $\lambda \in \mathbb{C}$ is an eigenvalue of $A$ when there exists a vector $\vec{x} \in \mathbb{C}^n$ with $\vec{x} \ne \vec{0}$ satisfying $A\vec{x} = \lambda\vec{x}$ . Any such $\vec{x}$ is an eigenvector of $A$ for $\lambda$ .

The exclusion $\vec{x} \ne \vec{0}$ is essential. The zero vector satisfies $A\vec{0} = \lambda\vec{0}$ for every scalar, so admitting it would make every complex number an eigenvalue of every matrix.

Characteristic Polynomial

Let $A$ be a square matrix of size $n$ . The characteristic polynomial of $A$ is the polynomial in the indeterminate $x$ given by $p_A(x) = \det\left(A - x I_n\right)$ , where $I_n$ is the identity matrix of size $n$ .

Some texts define the characteristic polynomial as $\det(xI_n - A)$ , which is $(-1)^n$ times the version used here. The two have identical roots, so eigenvalues are unaffected; only the sign of the leading coefficient changes.

Eigenspace of a Matrix

Let $A$ be a square matrix and let $\lambda$ be an eigenvalue of $A$ . The eigenspace of $A$ for $\lambda$ , written $\mathcal{E}_A(\lambda)$ , is the set containing every eigenvector of $A$ for $\lambda$ together with the zero vector.

The zero vector is included deliberately. It is never an eigenvector, but without it the set would fail to be closed under scalar multiplication by $0$ and so would not be a subspace.

Concepts

The chain of equivalences that makes computation possible

Four statements about a fixed scalar $\lambda$ are equivalent, and the chain between them is the whole computational strategy. There exists $\vec{x} \ne \vec{0}$ with $A\vec{x} = \lambda\vec{x}$ ; equivalently $A\vec{x} - \lambda I_n \vec{x} = \vec{0}$ ; equivalently $(A - \lambda I_n)\vec{x} = \vec{0}$ has a non-trivial solution; equivalently $A - \lambda I_n$ is singular; equivalently $\det(A - \lambda I_n) = 0$ . The insertion of $I_n$ in the second step is the only subtle move — $\lambda \vec{x}$ is a scalar multiple of a vector, and it must be rewritten as a matrix acting on a vector before the distributive law for matrix multiplication can factor $\vec{x}$ out.

Eigenvalues are exactly the roots of the characteristic polynomial

The chain above proves the central theorem of eigenvalue computation: for a square matrix $A$ , the scalar $\lambda$ is an eigenvalue of $A$ if and only if $p_A(\lambda) = 0$ . The practical force of the if and only if is that the polynomial finds all the eigenvalues and only the eigenvalues. There is no risk of a hidden eigenvalue that the polynomial fails to detect, and no root that turns out to be spurious. Combined with the fact that $p_A$ has degree $n$ , the spectrum of an $n \times n$ matrix is a finite set of at most $n$ complex numbers.

Why the eigenspace is a subspace

Closure is inherited directly from linearity. If $A\vec{x} = \lambda\vec{x}$ and $A\vec{y} = \lambda\vec{y}$ then $A(\vec{x} + \vec{y}) = \lambda(\vec{x} + \vec{y})$ , and $A(\alpha\vec{x}) = \alpha A\vec{x} = \lambda(\alpha\vec{x})$ . The sum or multiple could be the zero vector, which is not an eigenvector — and that is precisely the reason $\vec{0}$ is written into the definition of $\mathcal{E}_A(\lambda)$ . With it, the set is non-empty and closed under both operations, so it is a subspace of $\mathbb{C}^n$ and therefore has a well-defined dimension and a finite basis. Note carefully that this closure holds within a single eigenvalue; eigenvectors for two different eigenvalues do not add to give an eigenvector.

The eigenspace is a null space

For an eigenvalue $\lambda$ of a square matrix $A$ of size $n$ , $\mathcal{E}_A(\lambda) = \mathcal{N}(A - \lambda I_n)$ . The proof is the same chain of equivalences run for a fixed $\vec{x}$ instead of a fixed $\lambda$ , with the zero vector checked separately: $\vec{0}$ lies in the eigenspace by definition and in the null space because every homogeneous system admits the trivial solution. The consequence is operational. Computing an eigenspace requires no new algorithm at all — form the matrix $A - \lambda I_n$ , row-reduce it, and read a basis of the null space from the free variables in the usual way.

Reading a basis from the reduced form

Once $A - \lambda I_n$ is in reduced row-echelon form with $r$ pivot columns, the null space has dimension $n - r$ and a basis is produced by setting each free variable to $1$ in turn with the others zero. Because $\lambda$ was chosen to make the matrix singular, $r < n$ is guaranteed and at least one basis vector must appear; a computation that yields only the zero vector is an arithmetic error or a wrong root, not a legitimate outcome. Basis vectors may be scaled by any non-zero constant, so it is conventional to clear fractions and present the smallest integer representative.

The characteristic polynomial is a proof tool, not a production algorithm

Expanding $\det(A - xI_n)$ symbolically costs on the order of $n!$ operations by cofactor expansion, and the coefficients of the resulting polynomial are catastrophically ill-conditioned functions of the matrix entries: a relative perturbation of $10^{-16}$ in a coefficient can move a repeated root by $10^{-8}$ or worse. Production eigenvalue software never forms the characteristic polynomial. It reduces the matrix to Hessenberg form and runs the shifted QR iteration, which converges to the eigenvalues in roughly $25n^3$ operations with backward-stable error behaviour. The polynomial route remains the right tool for hand computation, for exact symbolic work, and for every theoretical argument in this part of the subject.

Procedure: eigenvalues and eigenvectors of a square matrix

Form $A - xI_n$ Subtract the indeterminate

x

from every diagonal entry of

A

, leaving the off-diagonal entries untouched.

Expand the determinantCompute

p_A(x) = \det(A - xI_n)

by cofactor expansion along the sparsest row or column. The result has degree

n

Factor and collect rootsSolve

p_A(x) = 0

over

\mathbb{C}

. The distinct roots are the eigenvalues; the exponent on each factor is its algebraic multiplicity.

For each eigenvalue, form $A - \lambda I_n$ Substitute the numeric root back into the matrix. This matrix must be singular — verify it, as a rank of

n

indicates an error upstream.

Row-reduce and extract the null spaceReduce

A - \lambda I_n

to reduced row-echelon form and write a basis for

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

from the free variables.

Normalise and verifyClear fractions in each basis vector, then confirm

A\vec{v} = \lambda\vec{v}

by direct multiplication before using the result.

Equations

Eigenvalue-eigenvector equation

EQ-CEE-01

Eigenvalue-eigenvector equation

The defining relation. The matrix acts on $\vec{x}$ by pure scaling: direction is preserved, only magnitude and sign change.

Homogeneous form of the eigenvalue equation

EQ-CEE-02

(A - λ I_{n}) \vec{x} = \vec{0}

Obtained by rewriting $\lambda\vec{x}$ as $\lambda I_n \vec{x}$ and factoring. This is the step that converts the problem into standard linear-system form.

Characteristic polynomial

EQ-CEE-03

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

A polynomial of degree $n$ in $x$ with leading coefficient $(-1)^n$ , constant term $\det(A)$ , and the coefficient of $x^{n-1}$ equal to $(-1)^{n-1}\operatorname{tr}(A)$ .

Eigenvalue criterion

EQ-CEE-04

λ is an eigenvalue of A \Leftrightarrow p_{A} (λ) = 0

The determinant test. It is an equivalence, so the roots of $p_A$ are precisely the eigenvalues — all of them, and nothing else.

Eigenspace as a null space

EQ-CEE-05

E_{A} (λ) = N (A - λ I_{n}) = {\vec{x} \in ℂ^{n} ∣ (A - λ I_{n}) \vec{x} = \vec{0}}

Every eigenvector for $\lambda$ , plus the zero vector. The identification makes the eigenspace a subspace with a computable basis.

Dimension of an eigenspace

EQ-CEE-06

\dim (E_{A} (λ)) = n - rank (A - λ I_{n}) \geq 1

The geometric multiplicity of $\lambda$ . It is at least $1$ because $A - \lambda I_n$ is singular, and at most $n$ .

Factored characteristic polynomial

EQ-CEE-07

p_{A} (x) = (- 1)^{n} (x - λ_{1})^{α_{1}} (x - λ_{2})^{α_{2}} \dots (x - λ_{k})^{α_{k}}, α_{1} + α_{2} + \dots + α_{k} = n

Complete factorisation over $\mathbb{C}$ . The distinct roots $\lambda_i$ are the eigenvalues and the exponents $\alpha_i$ are their algebraic multiplicities.

Variable Definitions

Symbols used on this page
Symbol	Name	Meaning	Domain / type
$A$	Matrix	The square matrix whose spectrum is sought	n x n complex matrix
$n$	Size	Number of rows, equal to the number of columns	positive integer
$λ$	Eigenvalue	Scalar for which $A - \lambda I_n$ is singular	complex number
$\vec{x}$	Eigenvector	Non-zero vector scaled by $\lambda$ under multiplication by $A$	non-zero vector in C^n
$I_{n}$	Identity matrix	Square matrix with ones on the diagonal and zeros elsewhere	n x n matrix
$p_{A} (x)$	Characteristic polynomial	$\det(A - xI_n)$ , whose roots are the eigenvalues	polynomial of degree n
$E_{A} (λ)$	Eigenspace	All eigenvectors for $\lambda$ together with $\vec{0}$	subspace of C^n
$N (M)$	Null space	Set of vectors annihilated by the matrix $M$	subspace of C^n
$rank (M)$	Rank	Number of pivot columns in the reduced row-echelon form of $M$	0 to n

Worked Numerical Example

Problem statement

A three-storey shear building is modelled by a symmetric stiffness-like matrix $K$ relating storey displacements to restoring forces. Find the natural modes of the model by computing every eigenvalue of $K$ and a basis for each eigenspace.

State the matrix
The assembled matrix couples each storey only to its immediate neighbours, so it is tridiagonal and symmetric:
$K = [\begin{matrix} 7 & - 2 & 0 \\ - 2 & 6 & - 2 \\ 0 & - 2 & 5 \end{matrix}]$
Form $K - xI_3$
Subtract $x$ from each diagonal entry. Nothing off the diagonal changes — a point worth stating explicitly, because subtracting $x$ from every entry is the single most common slip at this step.
$Form $K - xI_3$$
Expand the determinant along the first row
The zero in position $(1,3)$ kills one of the three cofactor terms, so only two products need evaluating. The cofactor of the $(1,1)$ entry is $(6-x)(5-x) - 4 = x^2 - 11x + 26$ ; the cofactor of the $(1,2)$ entry is $-\left[(-2)(5-x) - 0\right] = 10 - 2x$ .
$p_{K} (x) = (7 - x) (x^{2} - 11 x + 26) + (- 2) (10 - 2 x)$
Collect terms
Multiplying out gives $7x^2 - 77x + 182 - x^3 + 11x^2 - 26x$ from the first product and $-20 + 4x$ from the second. Summing and factoring:
$p_{K} (x) = - x^{3} + 18 x^{2} - 99 x + 162 = - (x - 3) (x - 6) (x - 9)$
Read off the eigenvalues
The roots are $\lambda = 3$ , $\lambda = 6$ and $\lambda = 9$ . All three are real and distinct, each with algebraic multiplicity $1$ , and their sum $3 + 6 + 9 = 18$ matches $\operatorname{tr}(K) = 7 + 6 + 5$ — a free arithmetic check that catches most sign errors immediately. Their product $162$ matches $\det(K)$ .
Compute the eigenspace for $\lambda = 3$
Form $K - 3I_3$ and reduce. The reduced form has two pivot columns, so the null space is one-dimensional; setting the free variable $x_3 = 1$ and clearing the halves gives the integer representative below.
$Compute the eigenspace for $\lambda = 3$$
Repeat for $\lambda = 6$ and $\lambda = 9$
The same reduction applied to $K - 6I_3$ and $K - 9I_3$ produces one free variable in each case, giving one-dimensional eigenspaces:
$Repeat for $\lambda = 6$ and $\lambda = 9$$
Verify by direct multiplication
Never trust a reduction without substituting back. For the first mode, $K\begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix} = \begin{bmatrix} 7 - 4 + 0 \\ -2 + 12 - 4 \\ 0 - 4 + 10 \end{bmatrix} = \begin{bmatrix} 3 \\ 6 \\ 6 \end{bmatrix} = 3\begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix}$ , as required. The other two verify identically.

Result

The model has three distinct real eigenvalues $3, 6, 9$ , each with a one-dimensional eigenspace. Physically these are three independent modes: the softest mode $(1,2,2)$ moves all storeys in phase with increasing amplitude up the building, while the stiffest mode $(2,-2,1)$ alternates in sign, the classic high-frequency shape. The three mode vectors are mutually orthogonal, which is not an accident — it follows from the symmetry of $K$ , and it means the modes decouple the response completely.

Applications & Industry Use

Structural engineering

Natural frequencies and mode shapes

Free vibration of a discretised structure reduces to the generalised problem $K\vec{\phi} = \omega^2 M \vec{\phi}$ . After a mass normalisation this becomes a standard eigenvalue problem whose eigenvalues are squared natural frequencies and whose eigenvectors are mode shapes. Design codes require the first few modes to be identified so that seismic and wind loading can be distributed across them.

Control systems

Pole placement and stability margins

The eigenvalues of the state matrix $A$ in $\dot{\vec{x}} = A\vec{x}$ are the system poles. A continuous-time model is asymptotically stable exactly when every eigenvalue has negative real part, and the eigenvector associated with the slowest pole shows which states dominate the settling transient. Controller design is largely the business of relocating these roots.

Chemical & process engineering

Reaction time-scale separation

A linearised reaction network has a Jacobian whose eigenvalues are reciprocal time constants. Widely separated eigenvalues signal a stiff system that demands an implicit integrator, and the eigenvectors identify which species combinations relax fast enough to be treated as quasi-steady, allowing a justified model reduction.

Computing & data

Principal component analysis

The covariance matrix of a data set is symmetric and positive semi-definite. Its eigenvalues rank the variance captured along each orthogonal direction and its eigenvectors give those directions, so retaining the largest few eigenpairs compresses the data with a quantified loss. The eigenvalue spectrum itself is the diagnostic used to choose how many components to keep.

Electrical power systems

Small-signal oscillation analysis

Interconnected generators are linearised about an operating point, and the eigenvalues of the resulting state matrix reveal inter-area oscillation modes. A lightly damped complex pair near the imaginary axis is a known precursor to instability, and the participation factors derived from the eigenvectors identify which machines to fit with damping controllers.

Mechanical engineering

Principal stresses and principal axes

The Cauchy stress tensor at a point is a symmetric $3 \times 3$ matrix. Its eigenvalues are the principal stresses and its eigenvectors are the principal directions, on which the shear components vanish. Failure criteria are stated in terms of these eigenvalues, so the eigenvalue computation is a routine step in a strength assessment.

Design Considerations

Use the characteristic polynomial only at small sizes

Hand expansion of $\det(A - xI_n)$ is comfortable for $n = 2$ and $n = 3$ , tolerable for $n = 4$ if the matrix is sparse or structured, and impractical beyond that. Above $n = 4$ the polynomial route is a source of arithmetic error rather than insight; switch to a numerical eigensolver or to structural arguments about the matrix.

Exploit structure before computing

A triangular or diagonal matrix has its eigenvalues on the diagonal with no work at all, because the determinant of a triangular matrix is the product of its diagonal entries. Block-triangular matrices contribute the union of the spectra of their diagonal blocks. Checking for such structure first can eliminate the computation entirely.

Verify the trace and determinant identities

The sum of the eigenvalues, counted with algebraic multiplicity, equals $\operatorname{tr}(A)$ ; their product equals $\det(A)$ . Both checks cost a few additions and catch the majority of sign and transcription errors. Applying them before computing eigenvectors avoids wasting a full row reduction on a wrong root.

Decide early whether real arithmetic is enough

A real matrix may have no real eigenvalue at all — a plane rotation is the standard case. If the model requires real modes, that requirement must come from a structural property such as symmetry, not from hope. Where complex eigenvalues appear, they arrive in conjugate pairs for a real matrix, and the pair should be interpreted as an oscillatory mode rather than as two independent behaviours.

Scale the matrix before a numerical solve

Eigenvalue accuracy degrades when rows and columns have wildly differing magnitudes, which typically happens when a model mixes physical units. Balancing — a diagonal similarity transformation that equalises row and column norms — is applied automatically by LAPACK drivers and should never be disabled without cause, since it costs almost nothing and can improve accuracy by orders of magnitude.

Distinguish an eigenvector from a normalised eigenvector

Eigenvectors are determined only up to a non-zero scalar multiple, so no library can return the eigenvector. Numerical routines return unit-norm vectors with an arbitrary phase or sign convention; symbolic routines return integer representatives. Any downstream code that compares eigenvectors across runs or across libraries must fix its own normalisation.

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 notation used here: upright $\det$ for the determinant operator, italic $\lambda$ for the eigenvalue as a scalar variable, and the bracket form for matrices.
`IEEE 754-2019`	IEEE Standard for Floating-Point Arithmetic	Defines the binary64 arithmetic in which numerical eigensolvers operate, and therefore the unit roundoff that bounds the achievable accuracy of a computed eigenvalue.
`LAPACK reference implementation`	Linear Algebra PACKage driver routines	The de facto standard interface for eigenvalue computation: `xGEEV` for general matrices, `xSYEV` and `xHEEV` for symmetric and Hermitian ones. None of these forms the characteristic polynomial.
`ISO 4866`	Mechanical vibration and shock — Vibration of fixed structures	Structural modal assessments reported under this standard rest on natural frequencies and mode shapes obtained as eigenvalues and eigenvectors of the discretised model.
`ISO/IEC 40314`	Mathematical Markup Language (MathML) Version 3.0	Encodes every matrix and equation on this page as semantic markup, keeping the notation searchable 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	Small integer or rational matrices where the characteristic polynomial is to be factored exactly and eigenspace bases must be certified.	Delivers exact multiplicities and exact bases, but requires the polynomial to factor over the rationals; irrational or complex roots force a switch to symbolic radicals or numerical approximation.
Algebraic numbers (symbolic radicals)	Degree three or four characteristic polynomials that do not factor rationally but must still be expressed in closed form.	Exact but rapidly unwieldy: nested radicals make subsequent eigenvector computation and any comparison of magnitudes expensive and hard to simplify.
IEEE 754 binary64 with a QR eigensolver	Any engineering matrix beyond about $4 \times 4$ , and every case where the entries carry measurement uncertainty.	Backward stable and fast, but eigenvalues of a non-normal matrix can be far more sensitive than the backward error suggests; report a condition estimate alongside the values.
IEEE 754 binary32	Large-scale or GPU-accelerated spectral computations where memory bandwidth dominates and only a coarse spectrum is needed.	Roughly seven significant decimal digits, which is often insufficient to separate closely spaced eigenvalues or to resolve near-defective behaviour.
Real symmetric storage (packed or banded)	The matrix is known to be symmetric or Hermitian, as for stiffness, covariance and stress matrices.	Halves storage and permits a specialised algorithm that is faster and more accurate, but silently gives wrong answers if the symmetry assumption is violated by the assembly code.
Sparse storage with an iterative (Krylov) eigensolver	Very large matrices from finite element or network models where only a few extreme eigenvalues are wanted.	Avoids ever forming a dense factorisation, but returns only a selected part of the spectrum and requires careful choice of shift, tolerance and restart strategy.

Manufacturing Notes

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

Hand procedure and its cost

For $n = 2$ the characteristic polynomial is $x^2 - \operatorname{tr}(A)x + \det(A)$ and can be written down by inspection. For $n = 3$ , expand along whichever row or column carries the most zeros; a single zero eliminates a third of the work. Each eigenspace then costs one row reduction of an $n \times n$ matrix, roughly $\tfrac{2}{3}n^3$ operations, which is negligible next to the polynomial expansion at these sizes.

What production eigensolvers actually do

The standard dense algorithm reduces $A$ to upper Hessenberg form by orthogonal similarity transformations, then applies the implicitly shifted QR iteration until the subdiagonal entries decouple the matrix into $1 \times 1$ and $2 \times 2$ blocks. Total cost is about $10n^3$ operations for eigenvalues alone and around $25n^3$ if eigenvectors are also required. For symmetric matrices the reduction is to tridiagonal form and the cost falls by roughly a factor of four.

Library behaviour and return conventions

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 bugs. numpy.linalg.eigh assumes symmetry, reads only one triangle, and returns eigenvalues in ascending order. sympy.Matrix.eigenvects returns exact triples of eigenvalue, algebraic multiplicity and a basis of the eigenspace, which is the closest match to the hand procedure described here.

Verification technique

Three independent checks are worth building into any pipeline. Compare $\sum \lambda_i$ against $\operatorname{tr}(A)$ and $\prod \lambda_i$ against $\det(A)$ . Compute the residual $\lVert A\vec{v} - \lambda\vec{v} \rVert$ relative to $\lVert A \rVert$ for each computed pair. For a symmetric matrix, confirm that eigenvectors for distinct eigenvalues are orthogonal to within roundoff.

Numerical hazards specific to the polynomial route

Forming the characteristic polynomial and then root-finding is unstable in a way that has nothing to do with the quality of the root-finder. A root of multiplicity $m$ is perturbed by roughly the $m$ -th root of the coefficient perturbation, so a double root computed in binary64 carries only about eight correct digits and a triple root about five. This is the technical reason the method is confined to exact arithmetic.

Failure Modes & Common Mistakes

Failure modes, root causes and prevention
Failure mode / mistake	Impact	Root cause	Prevention & detection
Subtracting $x$ from every entry	high	Writing $A - x$ instead of $A - xI_n$ , so off-diagonal entries are altered as well.	Always write the identity explicitly. The scalar $x$ must be multiplied by $I_n$ before it can be subtracted from a matrix at all.
Accepting the zero vector as an eigenvector	high	Solving $(A - \lambda I_n)\vec{x} = \vec{0}$ and reporting the trivial solution.	The definition excludes $\vec{0}$ . If the null space is trivial then $\lambda$ is not an eigenvalue and the root was wrong or the reduction was faulty.
Missing complex eigenvalues	high	Factoring the characteristic polynomial over $\mathbb{R}$ only and declaring an irreducible quadratic factor to have no roots.	Work over $\mathbb{C}$ . Every irreducible real quadratic contributes a conjugate pair of complex eigenvalues, and for a real matrix those pairs are always conjugate.
Sign error in the determinant expansion	medium	Omitting the alternating $(-1)^{i+j}$ cofactor sign, or mishandling the sign of $-x$ terms.	Check the computed polynomial against $\operatorname{tr}(A)$ and $\det(A)$ before proceeding to eigenvectors.
Treating the eigenspace as a set of vectors rather than a subspace	medium	Reporting only the basis vectors found by row reduction and forgetting that all their non-zero combinations are eigenvectors too.	State the result as a span. The basis is a description of an infinite set, not the set itself.
Assuming distinct eigenvalues	medium	Stopping after finding $k < n$ roots without checking multiplicities, or assuming each eigenspace is one-dimensional.	Factor the polynomial completely so the multiplicities sum to $n$ , then compute the dimension of each eigenspace independently.
Root-finding a numerically formed characteristic polynomial	high	Assembling the polynomial coefficients in floating point and passing them to a root solver.	Never do this. Use a QR-based eigensolver; the coefficient representation destroys accuracy irrecoverably, especially for repeated or clustered roots.
Applying eigenvalue methods to a non-square matrix	low	Attempting $A\vec{x} = \lambda\vec{x}$ when $A$ is $m \times n$ with $m \ne n$ , where the two sides have different sizes.	Eigenvalues are defined only for square matrices. For rectangular matrices the analogous object is the singular value decomposition.

FAQs

Why must the identity matrix be inserted when moving from

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

(A - \lambda I_n)\vec{x} = \vec{0}

The expression $A - \lambda$ is not defined: a matrix and a scalar cannot be subtracted. Writing $\lambda\vec{x}$ as $\lambda I_n \vec{x}$ turns the right-hand side into a matrix acting on $\vec{x}$ , and only then does the distributive law for matrix-vector products allow $\vec{x}$ to be factored out. Skipping this step produces the correct-looking answer for the wrong reason and leads directly to the error of subtracting $\lambda$ from every entry.

Can a matrix with all real entries have complex eigenvalues?

Yes, and it is common. A plane rotation through an angle that is not a multiple of $\pi$ maps no real direction to a multiple of itself, so it has no real eigenvalue at all; its characteristic polynomial is an irreducible real quadratic with a conjugate pair of complex roots. For a real matrix, complex eigenvalues always occur in conjugate pairs, and the corresponding eigenvectors are conjugates of one another.

Is the eigenspace for

\lambda

the same as the set of eigenvectors for

\lambda

Almost. The eigenspace is the set of eigenvectors together with the zero vector. That single extra element is what makes the set closed under scalar multiplication and hence a subspace, which in turn gives it a dimension and a basis. The zero vector itself is never an eigenvector.

How many eigenvalues can an

n \times n

matrix have?

At most $n$ distinct ones. The characteristic polynomial has degree exactly $n$ , so counted with algebraic multiplicity there are exactly $n$ roots in $\mathbb{C}$ , and distinct roots therefore number at most $n$ . There is always at least one, because a non-constant polynomial always has a complex root.

Why do numerical libraries avoid the characteristic polynomial?

Because the map from matrix entries to polynomial coefficients, and from coefficients to roots, is severely ill-conditioned. A relative perturbation of size $\varepsilon$ in a coefficient can shift a root of multiplicity $m$ by roughly $\varepsilon^{1/m}$ , so even a perfectly accurate root-finder returns poor eigenvalues. The QR iteration works directly on the matrix and is backward stable, which is why every production library uses it.

What does it mean if

\lambda = 0

turns out to be an eigenvalue?

It means $A - 0 I_n = A$ is singular, so the matrix has a non-trivial null space and is not invertible. The eigenspace for $\lambda = 0$ is exactly $\mathcal{N}(A)$ . Equivalently, $\det(A) = 0$ , which is visible as a zero constant term in the characteristic polynomial.

Do I need to normalise the eigenvectors I compute?

Not for correctness — any non-zero multiple of an eigenvector is an eigenvector for the same eigenvalue. Normalisation matters for presentation and for comparison. By hand, clear fractions and present the smallest integer representative; numerically, scale to unit norm and fix a sign or phase convention so results are reproducible across runs and libraries.

References

Beezer, R. A. A First Course in Linear Algebra, Version 0.70. University of Puget Sound, 2006. Section EE, Subsection CEE. Licensed under the GNU Free Documentation License v1.2.
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.
ISO 80000-2:2019, Quantities and units — Part 2: Mathematics. International Organization for Standardization.
IEEE 754-2019, IEEE Standard for Floating-Point Arithmetic. Institute of Electrical and Electronics Engineers.

AI Suggested Questions

Expand $\det(A - xI_3)$ symbolically for a general $3 \times 3$ matrix and show how the coefficients relate to the trace, the sum of principal minors and the determinant.
Construct a real $2 \times 2$ matrix with no real eigenvalue and explain geometrically why no direction is preserved.
Compare the operation counts of characteristic-polynomial root-finding and the shifted QR iteration for a dense $200 \times 200$ matrix.
Show me a matrix where the characteristic polynomial has a double root but the eigenspace is only one-dimensional, and explain what goes wrong.
How sensitive is a computed eigenvalue to perturbations in the matrix entries, and what condition number quantifies that sensitivity?
Why does a symmetric matrix always produce mutually orthogonal eigenvectors for distinct eigenvalues?

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