Properties of Eigenvalues and Eigenvectors

Q: Why are eigenvectors for distinct eigenvalues automatically independent?

Because a dependence relation among them can be attacked from two sides. Multiplying the relation by $A$ scales each term by its own eigenvalue; multiplying by one of the eigenvalues scales every term equally. Subtracting eliminates one term and leaves a relation whose coefficients each carry a factor $\lambda_i - \lambda_k$. Distinctness makes those factors non-zero, forcing every coefficient to vanish and contradicting the assumed dependence.

Q: If $A$ and $A^{t}$ have the same eigenvalues, do they have the same eigenvectors?

No, not in general. The equality of spectra follows from $\det(M) = \det(M^{t})$ applied to $A - xI_n$, which says nothing about eigenvectors. The eigenvectors of $A^{t}$ are the left eigenvectors of $A$ and generally point elsewhere. They coincide when $A$ is symmetric, and more broadly when $A$ is normal.

Q: Can I compute the eigenvalues of $AB$ from those of $A$ and $B$?

Not in general. There is no formula, and simple counterexamples exist with $2 \times 2$ matrices. What is true is that $AB$ and $BA$ share the same non-zero eigenvalues with the same multiplicities, and that if $A$ and $B$ commute they can be simultaneously triangularised, so their eigenvalues can be paired multiplicatively.

Q: Does the polynomial rule give all the eigenvalues of $q(A)$ or only some?

Over the complex numbers it gives all of them: the spectrum of $q(A)$ is exactly the image of the spectrum of $A$ under $q$. The elementary argument only proves the inclusion in one direction, which is enough for most engineering purposes, but the equality is a standard result and is what makes spectral shaping a reliable design technique.

Q: What happens to the eigenvalues when I add a multiple of the identity?

They shift by that multiple: $A + cI_n$ has eigenvalues $\lambda + c$ with unchanged eigenvectors. This is the degenerate case of the polynomial rule with $q(x) = x + c$, and it is the mechanism behind shift strategies in iterative eigensolvers, which move a target eigenvalue near zero to accelerate convergence.

Q: Is the spectral radius the same as the matrix norm?

No. The spectral radius is always at most any induced matrix norm, and the two coincide for the two-norm only when the matrix is normal. A non-normal matrix can have spectral radius below one and still exhibit large transient growth in $\lVert A^{s} \rVert$ before eventual decay, which is why transient analysis cannot rest on eigenvalues alone.

Q: Why does the power rule require $s \ge 0$?

Because $A^{s}$ for negative $s$ means a power of the inverse, which only exists when $A$ is non-singular. Under that additional hypothesis the rule does extend: $A^{-s}$ has eigenvalues $\lambda^{-s}$, combining the power rule with the inverse rule. Without non-singularity, negative powers are undefined.

Overview

Computing a spectrum from the characteristic polynomial is expensive. Computing it a second time for a closely related matrix is usually unnecessary, because the operations that engineers actually apply to matrices — scaling, raising to a power, forming a polynomial, inverting, transposing — move the eigenvalues in an entirely predictable way while leaving the eigenvectors alone. Each of the theorems below is proved by the same short argument: start from $A\vec{x} = \lambda\vec{x}$ , apply the operation to both sides, and read off the new eigenvalue.

The reason the eigenvectors survive is worth stating plainly. On the subspace spanned by a single eigenvector, the matrix acts as multiplication by a scalar. Any expression built from $A$ and the identity by addition and multiplication therefore acts on that same subspace as the corresponding expression in $\lambda$ . This is the conceptual content of the whole family of results, and it is why a polynomial in $A$ has eigenvalues given by the same polynomial evaluated at the eigenvalues of $A$ .

The boundaries of the technique matter as much as its reach. There is no theorem giving the eigenvalues of $A + B$ or $A B$ in terms of the eigenvalues of $A$ and $B$ separately. The obstruction is exactly that two unrelated matrices do not share eigenvectors, so the argument above has nothing to work with. Assuming otherwise is one of the most consequential errors in applied linear algebra, and it recurs in stability analysis of coupled subsystems.

Two results on this page stand apart from the operational family. Eigenvectors belonging to distinct eigenvalues are always linearly independent — a fact that requires a genuine induction and that underwrites every diagonalisation argument. And a matrix is singular precisely when zero is one of its eigenvalues, which slots the spectrum directly into the standing list of equivalent conditions for non-singularity.

Definition

Spectrum of a Matrix

SPEC

The spectrum of a square matrix $A$ , written $\sigma(A)$ , is the set of all its eigenvalues — equivalently the set of distinct roots of the characteristic polynomial $p_A(x) = \det(A - xI_n)$ . For an $n \times n$ matrix over $\mathbb{C}$ the spectrum is a non-empty set of at most $n$ complex numbers.

The spectrum records only which values occur, not how often. Multiplicity information is carried separately by $\alpha_A(\lambda)$ and $\gamma_A(\lambda)$ .

Polynomial of a Matrix

For a square matrix $A$ of size $n$ and a polynomial $q(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_m x^m$ , the matrix $q(A)$ is defined as $a_0 I_n + a_1 A + a_2 A^2 + \cdots + a_m A^m$ . The constant term becomes a multiple of the identity, since a scalar cannot be added to a matrix.

Powers are defined by repeated multiplication with $A^0 = I_n$ . Because every term is a power of the same matrix, all terms commute with one another.

Concepts

Eigenvectors for distinct eigenvalues are linearly independent

Let $A$ be a square matrix and let $S = \{\vec{x}_1, \vec{x}_2, \ldots, \vec{x}_p\}$ be eigenvectors of $A$ for eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_p$ that are pairwise distinct. Then $S$ is linearly independent. The proof takes the shortest dependent initial segment $\{\vec{x}_1, \ldots, \vec{x}_k\}$ and writes a non-trivial relation $a_1\vec{x}_1 + \cdots + a_k\vec{x}_k = \vec{0}$ . Multiplying by $A$ replaces each $\vec{x}_i$ by $\lambda_i \vec{x}_i$ ; multiplying instead by the scalar $\lambda_k$ replaces each by $\lambda_k \vec{x}_i$ . Subtracting removes the $k$ -th term entirely and leaves $a_i(\lambda_i - \lambda_k) = 0$ for $i < k$ on a set already known to be independent. Distinctness forces $\lambda_i - \lambda_k \ne 0$ , hence every $a_i = 0$ , and the surviving relation $a_k \vec{x}_k = \vec{0}$ forces $a_k = 0$ because eigenvectors are non-zero. The relation was not non-trivial after all.

Zero eigenvalues detect singularity

A square matrix $A$ is singular if and only if $\lambda = 0$ is an eigenvalue of $A$ . The equivalence is immediate once written out: $A$ is singular exactly when some $\vec{x} \ne \vec{0}$ satisfies $A\vec{x} = \vec{0}$ , and $\vec{0} = 0\vec{x}$ , which is the eigenvalue equation with $\lambda = 0$ . Taking the contrapositive adds a new entry to the standing list of conditions equivalent to non-singularity: $A$ is non-singular if and only if $0 \notin \sigma(A)$ . This is the reason an eigenvalue computation doubles as an invertibility test, and why the eigenspace for $\lambda = 0$ is exactly $\mathcal{N}(A)$ .

Scalar multiples and powers

If $A\vec{x} = \lambda\vec{x}$ then $(\alpha A)\vec{x} = \alpha(A\vec{x}) = (\alpha\lambda)\vec{x}$ , so $\alpha\lambda$ is an eigenvalue of $\alpha A$ with the same eigenvector. Powers follow by induction: assuming $A^s \vec{x} = \lambda^s \vec{x}$ , applying $A$ once more gives $A^{s+1}\vec{x} = A^s(\lambda \vec{x}) = \lambda(A^s\vec{x}) = \lambda^{s+1}\vec{x}$ , with the base case $s = 0$ reading $I_n\vec{x} = \lambda^0 \vec{x}$ . So $\lambda^s$ is an eigenvalue of $A^s$ for every integer $s \ge 0$ . This is the analytical basis for reasoning about long-run behaviour: $\lVert A^s \rVert$ grows or decays according to the largest $\lvert \lambda \rvert$ , the spectral radius.

Polynomials of a matrix and spectral shaping

Combining the previous two results term by term gives the strongest statement of the family: if $\lambda$ is an eigenvalue of $A$ and $q$ is any polynomial, then $q(\lambda)$ is an eigenvalue of $q(A)$ , and $\vec{x}$ remains an eigenvector. Over $\mathbb{C}$ the converse also holds, so $\sigma(q(A))$ is exactly the image $q(\sigma(A))$ — the spectral mapping theorem. The engineering consequence is constructive: to obtain a matrix with a prescribed spectrum, build an interpolating polynomial that sends each existing eigenvalue to its target and apply it to $A$ . Multiplicities are not preserved by this construction, because $q$ may map two distinct eigenvalues to the same value.

Inverses and transposes

If $A$ is non-singular with eigenvalue $\lambda$ , then $\lambda \ne 0$ (otherwise the matrix would be singular) and $A^{-1}\vec{x} = \tfrac{1}{\lambda}\vec{x}$ , so the inverse has the reciprocal eigenvalues with identical eigenvectors. The transpose behaves differently. From $\det(M) = \det(M^{t})$ applied to $M = A - xI_n$ it follows that $p_A(x) = p_{A^{t}}(x)$ , so $A$ and $A^{t}$ have identical characteristic polynomials, identical eigenvalues and identical algebraic multiplicities. Their eigenvectors are generally different: the eigenvectors of $A^{t}$ are the left eigenvectors of $A$ , and confusing the two is a common and silent error in modal analysis of non-symmetric systems.

Real matrices produce conjugate pairs

Suppose $A$ has real entries and $A\vec{x} = \lambda\vec{x}$ . Conjugating every entry of both sides gives $\overline{A}\,\overline{\vec{x}} = \overline{\lambda}\,\overline{\vec{x}}$ , and $\overline{A} = A$ because the entries are real, so $\overline{\vec{x}}$ is an eigenvector of $A$ for $\overline{\lambda}$ . Complex eigenvalues of a real matrix therefore arrive in conjugate pairs with conjugate eigenvectors, and computing one member of a pair determines the other with no further work. The same conclusion follows from the characteristic polynomial having real coefficients, but the direct argument also delivers the eigenvector relationship.

What does not transfer

There is no theorem of the form $\sigma(A + B) = \sigma(A) + \sigma(B)$ or $\sigma(AB) = \sigma(A)\sigma(B)$ for general matrices, and both statements are false in easily constructed examples. The proofs above work only because every matrix involved is built from a single $A$ and therefore shares its eigenvectors. Two matrices with a common eigenvector do inherit the additive and multiplicative rule on that vector, which is why simultaneously diagonalisable families — equivalently, commuting families — are the exception that recovers the intuition.

Deriving a related spectrum without recomputation

Compute $\sigma(A)$ onceObtain the eigenvalues and eigenvectors of the base matrix by the characteristic polynomial or a numerical solver.

Is the target built from $A$ alone?Scalar multiples, powers, polynomials, inverses and transposes qualify. Sums or products with an unrelated matrix do not — recompute from scratch in that case.

Apply the corresponding scalar operation

\alpha A \to \alpha\lambda

;

A^s \to \lambda^s

;

q(A) \to q(\lambda)

;

A^{-1} \to 1/\lambda

when

A

is non-singular.

Carry the eigenvectors across unchangedEvery case above preserves eigenvectors exactly — except the transpose, whose eigenvectors are the left eigenvectors of

A

Re-derive the multiplicitiesA polynomial can merge distinct eigenvalues, so algebraic multiplicities must be recounted on the image spectrum rather than copied.

Equations

Independence of eigenvectors for distinct eigenvalues

EQ-PEE-01

λ_{i} \neq λ_{j} for i \neq j \Rightarrow {{\vec{x}}_{1}, {\vec{x}}_{2}, \dots, {\vec{x}}_{p}} is linearly independent

Eigenvectors chosen one per distinct eigenvalue are always independent, regardless of the size or structure of the matrix.

Singularity and the zero eigenvalue

EQ-PEE-02

A is singular \Leftrightarrow 0 \in σ (A) \Leftrightarrow \det (A) = 0

The eigenvalue $0$ , a trivial null space and a vanishing determinant are three descriptions of the same condition.

Scalar multiple and integer powers

EQ-PEE-03

A \vec{x} = λ \vec{x} \Rightarrow (α A) \vec{x} = (α λ) \vec{x}, A^{s} \vec{x} = λ^{s} \vec{x} (s \geq 0)

Both operations leave the eigenvector untouched and transform the eigenvalue by the same scalar operation.

Polynomial of a matrix

EQ-PEE-04

q (A) = a_{0} I_{n} + a_{1} A + a_{2} A^{2} + \dots + a_{m} A^{m}, q (A) \vec{x} = q (λ) \vec{x}

The spectral mapping rule. Building $q$ as an interpolating polynomial through prescribed target values produces a matrix with a designed spectrum.

Inverse and transpose

EQ-PEE-05

A^{- 1} \vec{x} = \frac{1}{λ} \vec{x} (A non-singular), p_{A^{t}} (x) = p_{A} (x)

The inverse has reciprocal eigenvalues with the same eigenvectors; the transpose has identical eigenvalues and multiplicities but different eigenvectors.

Conjugate pairs for a real matrix

EQ-PEE-06

\bar{A} = A, A \vec{x} = λ \vec{x} \Rightarrow A \bar{\vec{x}} = \bar{λ} \bar{\vec{x}}

Conjugating the eigenvalue equation of a real matrix produces a second, genuinely distinct eigenpair whenever $\lambda$ is not real.

Spectral radius and asymptotic growth

EQ-PEE-07

ρ (A) = max {| λ | : λ \in σ (A)}, A^{s} \to 0 \Leftrightarrow ρ (A) < 1

A direct consequence of the power rule: the largest eigenvalue modulus decides whether repeated application of $A$ decays or grows.

Variable Definitions

Symbols used on this page
Symbol	Name	Meaning	Domain / type
$A$	Matrix	The square matrix whose spectrum is known	n x n complex matrix
$λ$	Eigenvalue	An element of the spectrum of $A$	complex number
$\vec{x}$	Eigenvector	Non-zero vector satisfying $A\vec{x} = \lambda\vec{x}$	non-zero vector in C^n
$α$	Scalar factor	Multiplier applied to the whole matrix	complex number
$s$	Power index	Exponent in $A^{s}$ , with $A^{0} = I_n$	non-negative integer
$q (x)$	Polynomial	Polynomial applied to the matrix to reshape its spectrum	polynomial over C
$σ (A)$	Spectrum	Set of distinct eigenvalues of $A$	subset of C
$ρ (A)$	Spectral radius	Largest modulus among the eigenvalues	non-negative real
$A^{t}$	Transpose	Matrix with rows and columns interchanged	n x n matrix

Worked Numerical Example

Problem statement

A discrete-time process model has state matrix $K$ with known spectrum. A designer proposes replacing the plant model by a shaped version $q(K)$ intended to place two of the three modes at the origin. Derive the spectra of $2K$ , $K^2$ , $K^{-1}$ , $K^{t}$ and $q(K)$ without recomputing any characteristic polynomial.

Start from the known spectrum
The base matrix is symmetric and tridiagonal, with eigenvalues and eigenvectors established by direct computation:
$K = [\begin{matrix} 7 & - 2 & 0 \\ - 2 & 6 & - 2 \\ 0 & - 2 & 5 \end{matrix}], σ (K) = {3, 6, 9}$
Scalar multiple
Doubling the matrix doubles every eigenvalue and changes no eigenvector. The check is immediate: $\operatorname{tr}(2K) = 36 = 6 + 12 + 18$ .
$σ (2 K) = {6, 12, 18}$
Square of the matrix
Squaring squares each eigenvalue. Since $\rho(K) = 9 > 1$ , repeated application of $K$ grows without bound — a discrete-time model with this state matrix is unstable, and no amount of eigenvector analysis changes that conclusion.
$K^{2} = [\begin{matrix} 53 & - 26 & 4 \\ - 26 & 44 & - 22 \\ 4 & - 22 & 29 \end{matrix}], σ (K^{2}) = {9, 36, 81}$
Inverse
No eigenvalue is zero, so $K$ is non-singular and the inverse exists. Its eigenvalues are the reciprocals, with the same eigenvectors. Their product $\tfrac{1}{3}\cdot\tfrac{1}{6}\cdot\tfrac{1}{9} = \tfrac{1}{162}$ equals $\det(K^{-1}) = 1/162$ , confirming the arithmetic.
$σ (K^{- 1}) = {\frac{1}{3}, \frac{1}{6}, \frac{1}{9}}$
Transpose
Here $K$ is symmetric, so $K^{t} = K$ and the spectra coincide trivially. For a non-symmetric matrix the eigenvalues would still coincide, but the eigenvectors of $K^{t}$ would be the left eigenvectors of $K$ and would generally point in different directions.
$σ (K^{t}) = σ (K) = {3, 6, 9}$
Construct the shaping polynomial
The design goal is to send $3 \mapsto 0$ and $9 \mapsto 0$ while leaving the middle mode non-zero. A polynomial with roots at $3$ and $9$ does exactly this, and its value at $6$ follows automatically.
$q (x) = (x - 3) (x - 9) = x^{2} - 12 x + 27, q (3) = 0, q (6) = - 9, q (9) = 0$
Form $q(K)$ and read its spectrum
Substituting the matrix for the indeterminate, with the constant term becoming $27 I_3$ , gives a rank-one matrix. The spectral mapping rule predicts eigenvalues $\{0, -9, 0\}$ , so $\lambda = 0$ now has algebraic multiplicity $2$ — the polynomial has merged two originally distinct eigenvalues.
$Form $q(K)$ and read its spectrum$
Verify the eigenvector claim
The eigenvector of $K$ for $\lambda = 6$ is $(-2,-1,2)$ . Multiplying, $q(K)\begin{bmatrix} -2 \\ -1 \\ 2 \end{bmatrix} = \begin{bmatrix} 8 + 2 + 8 \\ 4 + 1 + 4 \\ -8 - 2 - 8 \end{bmatrix} = \begin{bmatrix} 18 \\ 9 \\ -18 \end{bmatrix} = -9\begin{bmatrix} -2 \\ -1 \\ 2 \end{bmatrix}$ . The eigenvector is unchanged and the eigenvalue is $q(6) = -9$ , exactly as predicted.

Result

Five related spectra were obtained from one computation. The shaped matrix $q(K)$ is singular by construction, since zero entered its spectrum the moment the polynomial was given roots at two existing eigenvalues; its rank is $1$ and its null space is the plane spanned by the eigenvectors for $3$ and $9$ . The example also exposes the limit of the technique: multiplicities are not preserved, so the merged eigenvalue at zero must have its geometric multiplicity checked independently rather than inherited.

Applications & Industry Use

Control systems

Stability from the spectral radius

A discrete-time model $\vec{x}_{k+1} = A\vec{x}_k$ decays to the origin from every initial state exactly when every eigenvalue has modulus less than one. The power rule is what licenses that statement: on each eigenvector the state is multiplied by $\lambda^k$ , and the slowest-decaying mode is the one with the largest $\lvert \lambda \rvert$ .

Structural engineering

Reciprocal spectra in flexibility and stiffness

Flexibility is the inverse of stiffness, so the flexibility matrix has the reciprocal eigenvalues of the stiffness matrix with identical mode shapes. The softest structural mode is therefore the stiffest flexibility mode, and an analysis performed in one formulation transfers to the other without recomputation.

Computing & data

Convergence of iterative solvers

Stationary iterations such as Jacobi and Gauss-Seidel converge if and only if the iteration matrix has spectral radius below one, and the asymptotic rate is governed by that value. Preconditioning is precisely the practice of replacing the iteration matrix by a related one, often a polynomial in the original, with a smaller spectral radius.

Chemical & process engineering

Zero eigenvalues as conservation laws

A zero eigenvalue in a linearised network model is equivalent to singularity of the system matrix, and the associated eigenvector identifies a conserved combination of species. Detecting it through the spectrum rather than through a failed matrix inversion gives the diagnosis and the conserved quantity in one step.

Electrical engineering

Left eigenvectors and participation factors

Modal analysis of a power network uses the eigenvectors of $A$ to describe mode shape and the eigenvectors of $A^{t}$ to describe mode observability. Because both matrices share a spectrum but not their eigenvectors, the two sets must be computed and labelled separately; combining them elementwise gives the participation factors used to site damping controllers.

Aerospace engineering

Designed spectra by polynomial shaping

Where a plant matrix must be modified to place modes at target locations without changing the physical eigenvectors, an interpolating polynomial applied to the matrix achieves it directly. The construction guarantees the intended eigenvalue placement while preserving every mode shape, which is valuable when the mode shapes have physical meaning that must be retained.

Design Considerations

Never assume eigenvalues add or multiply across matrices

The eigenvalues of $A + B$ bear no simple relation to those of $A$ and $B$ unless the two matrices share a full set of eigenvectors, which is equivalent to their commuting. Coupled subsystems almost never commute, so the stability of an interconnection cannot be inferred from the stability of its parts. Compute the spectrum of the combined matrix.

Distinguish right from left eigenvectors

$A$ and $A^{t}$ have the same eigenvalues, and this equality tempts practitioners to treat their eigenvectors as interchangeable. They are not, except when the matrix is symmetric. Any code that assembles a modal transformation must be explicit about which side it is using, and any documentation should name the convention.

Check invertibility through the spectrum when it is already available

If eigenvalues have been computed for another purpose, invertibility is free: the matrix is non-singular exactly when no eigenvalue is zero, and the condition number in the two-norm of a normal matrix is the ratio of largest to smallest eigenvalue modulus. For non-normal matrices this ratio is a lower bound, not the condition number, and singular values must be used instead.

Recount multiplicities after a spectral mapping

A polynomial can send two distinct eigenvalues to the same image, merging their algebraic multiplicities, and it can send an eigenvalue to zero, making a non-singular matrix singular. The eigenvectors carry across, but the multiplicity bookkeeping does not. Recount on the image spectrum.

Beware inverting near-zero eigenvalues numerically

The theorem that $A^{-1}$ has eigenvalues $1/\lambda$ is exact, but it also explains why inversion amplifies error: an eigenvalue computed with absolute error $\varepsilon$ near zero produces a reciprocal with error of order $\varepsilon / \lambda^2$ . Where the smallest eigenvalue is close to the noise floor, solve a linear system rather than forming an inverse.

Use the power rule for asymptotics, not for computation

Knowing that $A^s$ has eigenvalues $\lambda^s$ answers questions about growth and decay, but computing $A^s$ by repeated multiplication is both expensive and numerically poor for large $s$ . Use a diagonalisation or a scaling-and-squaring scheme, and reserve the eigenvalue rule for the qualitative conclusion.

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	Sets the conventions for the conjugate bar, the transpose superscript and the use of upright operator names, all of which appear throughout the theorems on this page.
`IEC 61508`	Functional safety of electrical/electronic/programmable electronic safety-related systems	Stability arguments submitted in a safety case for a digital controller rest on the spectral radius of the closed-loop state matrix, which is the power rule applied to the discretised model.
`LAPACK reference implementation`	Linear Algebra PACKage driver routines	`xGEEV` optionally returns left eigenvectors alongside right ones precisely because the transpose shares eigenvalues but not eigenvectors; the two output arrays must not be confused.
`IEEE 754-2019`	IEEE Standard for Floating-Point Arithmetic	Governs the accuracy of reciprocal eigenvalues and of repeated squaring, where the standard's rounding guarantees bound the error growth in the power rule.
`ISO/IEC 40314`	Mathematical Markup Language (MathML) Version 3.0	Encodes the set-valued spectral notation and the conjugate and transpose accents used here as semantic markup rather than images.

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 for polynomial shaping	Constructing an interpolating polynomial to place a spectrum, where the coefficients must be exact for the target values to be hit precisely.	Guarantees the designed eigenvalues exactly, but the coefficients of a high-degree interpolant grow large and the resulting matrix polynomial becomes expensive to form.
IEEE 754 binary64	Routine engineering spectra, stability margins and spectral radius estimates.	Adequate for all qualitative conclusions, but reciprocals of small eigenvalues and high powers both amplify rounding error substantially.
Complex binary64 pairs	Real matrices with oscillatory modes, where conjugate pairs must be stored and manipulated.	Doubles storage per eigenvalue but keeps the conjugate structure explicit; halving the computation by conjugation recovers most of the cost.
Symmetric or Hermitian storage	The matrix equals its own transpose or adjoint, as for stiffness, covariance and Gram matrices.	Left and right eigenvectors coincide, eliminating an entire class of error, and the eigenvalues are guaranteed real; the assumption must be enforced at assembly time.
Scaled fixed-point	Embedded controllers where the stability test $\rho(A) < 1$ must be evaluated on a device without floating-point hardware.	Deterministic timing and small footprint, but the limited dynamic range makes eigenvalues near the unit circle indistinguishable from those on it.
Interval arithmetic	A certified bound on the spectral radius is needed rather than an estimate.	Produces guaranteed enclosures suitable for a formal argument, at a large runtime cost and with intervals that widen quickly under repeated squaring.

Manufacturing Notes

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

Cost of deriving a related spectrum

Every rule on this page is free in the computational sense: the derived eigenvalues cost $O(n)$ scalar operations once the base spectrum is known, against $O(n^3)$ for a fresh eigendecomposition. For a matrix polynomial the only real expense is forming $q(A)$ itself, which requires $m - 1$ matrix multiplications at $2n^3$ operations each, or fewer using Horner's scheme.

Forming a matrix polynomial correctly

Evaluate $q(A)$ by Horner's rule in matrix form: start from $a_m I_n$ , then repeatedly multiply by $A$ and add $a_k I_n$ . This uses $m-1$ matrix products rather than computing each power separately, and it never requires storing more than two intermediate matrices. Remember that the constant term becomes $a_0 I_n$ ; adding a bare scalar to a matrix is undefined.

Library behaviour for left eigenvectors

scipy.linalg.eig accepts left=True and right=True flags and returns the left eigenvectors satisfying

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

, that is the eigenvectors of the conjugate transpose rather than the plain transpose. NumPy's numpy.linalg.eig returns right eigenvectors only; obtaining left eigenvectors requires an explicit call on the transposed matrix.

Verification of a derived spectrum

Each derived spectrum admits its own cheap check. Compare the sum of the derived eigenvalues against the trace of the derived matrix and their product against its determinant. For the inverse, confirm that the product of all eigenvalues of $A$ and of $A^{-1}$ is $1$ . For a polynomial mapping, verify one eigenpair by direct multiplication as in the worked example.

Numerical caution with high powers

Computing $A^{s}$ by repeated multiplication accumulates error proportional to $s$ and can overflow when $\rho(A) > 1$ or underflow when $\rho(A) < 1$ . Scaling-and-squaring reduces the operation count to $O(\log s)$ products and is the standard approach; where only the asymptotic behaviour matters, use the spectral radius and do not form the power at all.

Failure Modes & Common Mistakes

Failure modes, root causes and prevention
Failure mode / mistake	Impact	Root cause	Prevention & detection
Assuming $\sigma(A + B) = \sigma(A) + \sigma(B)$	high	Extending the scalar-multiple and power rules to two unrelated matrices.	The rules hold only for expressions built from a single matrix. For a sum or product of distinct matrices, compute the spectrum of the combined matrix directly.
Using eigenvectors of $A^{t}$ as eigenvectors of $A$	high	Correctly noting that the two matrices share a spectrum and incorrectly assuming they share eigenvectors.	Compute right and left eigenvectors separately and label them. They coincide only when the matrix is symmetric or, more generally, normal.
Inverting a matrix with a zero eigenvalue	high	Applying the reciprocal rule without first confirming non-singularity.	Check that $0 \notin \sigma(A)$ before using $1/\lambda$ . A zero eigenvalue is exactly the condition under which the inverse does not exist.
Copying multiplicities through a polynomial map	medium	Assuming $\alpha_{q(A)}(q(\lambda)) = \alpha_A(\lambda)$ when $q$ merges two eigenvalues.	Recount algebraic multiplicities on the image spectrum, and compute geometric multiplicities from the null space of $q(A) - q(\lambda)I_n$ .
Adding a scalar to a matrix in a polynomial	medium	Writing $q(A) = A^2 + 3A + 5$ instead of $A^2 + 3A + 5I_n$ .	The constant term of a matrix polynomial is always a multiple of the identity. Most libraries will broadcast the scalar to every entry instead and produce a silently wrong matrix.
Concluding independence from repeated eigenvalues	medium	Applying the distinct-eigenvalue independence theorem to a set of eigenvectors that includes two for the same eigenvalue.	The theorem covers one eigenvector per distinct eigenvalue. Within a single eigenspace, independence must be established from the null space basis itself.
Reading the condition number off the eigenvalues	medium	Taking $\lvert\lambda_{\max}\rvert / \lvert\lambda_{\min}\rvert$ as the two-norm condition number of a non-normal matrix.	That ratio equals the condition number only for normal matrices. Otherwise use singular values, which can differ from eigenvalue moduli by orders of magnitude.
Expecting real eigenvalues from a real matrix	low	Ignoring that conjugate pairs are the normal outcome for a non-symmetric real matrix.	Work over $\mathbb{C}$ throughout, and require symmetry or another structural property if real eigenvalues are needed.

FAQs

Why are eigenvectors for distinct eigenvalues automatically independent?

Because a dependence relation among them can be attacked from two sides. Multiplying the relation by $A$ scales each term by its own eigenvalue; multiplying by one of the eigenvalues scales every term equally. Subtracting eliminates one term and leaves a relation whose coefficients each carry a factor $\lambda_i - \lambda_k$ . Distinctness makes those factors non-zero, forcing every coefficient to vanish and contradicting the assumed dependence.

A

and

A^{t}

have the same eigenvalues, do they have the same eigenvectors?

No, not in general. The equality of spectra follows from $\det(M) = \det(M^{t})$ applied to $A - xI_n$ , which says nothing about eigenvectors. The eigenvectors of $A^{t}$ are the left eigenvectors of $A$ and generally point elsewhere. They coincide when $A$ is symmetric, and more broadly when $A$ is normal.

Can I compute the eigenvalues of

A B

from those of

A

and

B

Not in general. There is no formula, and simple counterexamples exist with $2 \times 2$ matrices. What is true is that $A B$ and $B A$ share the same non-zero eigenvalues with the same multiplicities, and that if $A$ and $B$ commute they can be simultaneously triangularised, so their eigenvalues can be paired multiplicatively.

Does the polynomial rule give all the eigenvalues of

q(A)

or only some?

Over the complex numbers it gives all of them: the spectrum of $q(A)$ is exactly the image of the spectrum of $A$ under $q$ . The elementary argument only proves the inclusion in one direction, which is enough for most engineering purposes, but the equality is a standard result and is what makes spectral shaping a reliable design technique.

What happens to the eigenvalues when I add a multiple of the identity?

They shift by that multiple: $A + cI_n$ has eigenvalues $\lambda + c$ with unchanged eigenvectors. This is the degenerate case of the polynomial rule with $q(x) = x + c$ , and it is the mechanism behind shift strategies in iterative eigensolvers, which move a target eigenvalue near zero to accelerate convergence.

Is the spectral radius the same as the matrix norm?

No. The spectral radius is always at most any induced matrix norm, and the two coincide for the two-norm only when the matrix is normal. A non-normal matrix can have spectral radius below one and still exhibit large transient growth in $\lVert A^{s} \rVert$ before eventual decay, which is why transient analysis cannot rest on eigenvalues alone.

Why does the power rule require

s \ge 0

Because $A^{s}$ for negative $s$ means a power of the inverse, which only exists when $A$ is non-singular. Under that additional hypothesis the rule does extend: $A^{-s}$ has eigenvalues $\lambda^{-s}$ , combining the power rule with the inverse rule. Without non-singularity, negative powers are undefined.

References

Beezer, R. A. A First Course in Linear Algebra, Version 0.70. University of Puget Sound, 2006. Section PEE. Licensed under the GNU Free Documentation License v1.2.
Horn, R. A. and Johnson, C. R. Matrix Analysis, 2nd edition. Cambridge University Press, 2012.
Trefethen, L. N. and Embree, M. Spectra and Pseudospectra. Princeton University Press, 2005.
ISO 80000-2:2019, Quantities and units — Part 2: Mathematics. International Organization for Standardization.
Anderson, E. et al. LAPACK Users' Guide, 3rd edition. Society for Industrial and Applied Mathematics, 1999.

AI Suggested Questions

Give a concrete pair of $2 \times 2$ matrices where the eigenvalues of $A + B$ are not sums of eigenvalues of $A$ and $B$ .
Construct an interpolating polynomial that maps a given set of three eigenvalues to three prescribed targets, and show the resulting matrix.
Prove that $A B$ and $B A$ have the same non-zero eigenvalues with the same algebraic multiplicities.
Show a non-normal matrix with spectral radius below one whose powers grow substantially before decaying, and quantify the transient.
How do the left and right eigenvectors of a non-symmetric matrix combine to give participation factors in modal analysis?
Under what conditions on $A$ and $B$ can the eigenvalues of the product be obtained by pairing the eigenvalues of the factors?

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