Eigenvalues of Hermitian Matrices

Overview

Most matrices arising in engineering are not arbitrary. Stiffness and mass matrices, covariance matrices, Gram matrices, graph Laplacians and quantum observables all satisfy the same structural condition: the matrix equals its own conjugate transpose. Such a matrix is Hermitian, or self-adjoint, and when its entries are real the condition reduces to plain symmetry, $A = A^{t}$ .

Two theorems convert this structural condition into guarantees that no general matrix enjoys. The first is that every eigenvalue of a Hermitian matrix is real, even though eigenvalues in general live in $\mathbb{C}$ and even though a real non-symmetric matrix routinely produces complex conjugate pairs. The second is that eigenvectors belonging to different eigenvalues are not merely linearly independent but genuinely orthogonal — their inner product is zero.

Both proofs run through the inner product rather than through the characteristic polynomial, and both hinge on a single move: for a Hermitian matrix, $\langle A\vec{x}, \vec{y} \rangle = \langle \vec{x}, A\vec{y} \rangle$ . The matrix can be shifted from one argument of the inner product to the other without cost. Evaluating the same scalar with the matrix on the left and on the right produces the same number expressed two ways, and comparing the two expressions is the whole argument.

The engineering consequence is that a Hermitian model admits an orthonormal basis of eigenvectors. Modes decouple exactly, the transformation into modal coordinates is an orthogonal or unitary matrix whose inverse is its adjoint, and the numerical problem is perfectly conditioned in the sense that eigenvalue perturbations are bounded by the perturbation of the matrix itself. None of these properties survives if the symmetry is broken, which is why preserving it through assembly and discretisation is a design objective in its own right.

Definition

Adjoint of a Matrix

The adjoint of a matrix $A$ , written $A^{*}$ , is the conjugate transpose: $A^{*} = \overline{A}^{\,t}$ , so $[A^{*}]_{ij} = \overline{[A]_{ji}}$ . Transposing and conjugating commute, so the two operations may be applied in either order.

For a matrix with real entries the conjugation is inert and the adjoint is the ordinary transpose.

Hermitian Matrix

A square matrix $A$ is Hermitian, or self-adjoint, when $A = A^{*}$ . Entry by entry the condition reads $[A]_{ij} = \overline{[A]_{ji}}$ for all $i$ and $j$ .

Setting $i = j$ gives $[A]_{ii} = \overline{[A]_{ii}}$ , so every diagonal entry of a Hermitian matrix is real. A real Hermitian matrix is exactly a symmetric matrix.

Orthogonal Vectors

Two vectors $\vec{x}, \vec{y} \in \mathbb{C}^n$ are orthogonal when their inner product vanishes, $\langle \vec{x}, \vec{y} \rangle = 0$ , where $\langle \vec{u}, \vec{v} \rangle = \sum_{i=1}^{n} [\vec{u}]_i \overline{[\vec{v}]_i}$ .

Orthogonality is symmetric because the inner product obeys $\langle \vec{u}, \vec{v} \rangle = \overline{\langle \vec{v}, \vec{u} \rangle}$ , and a set of non-zero mutually orthogonal vectors is automatically linearly independent.

Concepts

The self-adjoint shift

The inner product on $\mathbb{C}^n$ can be written in matrix form as $\langle \vec{u}, \vec{v} \rangle = \vec{v}^{*}\vec{u}$ . Substituting $A\vec{u}$ for the first argument gives $\langle A\vec{u}, \vec{v} \rangle = \vec{v}^{*}A\vec{u} = \left(A^{*}\vec{v}\right)^{*}\vec{u} = \langle \vec{u}, A^{*}\vec{v} \rangle$ for any square matrix. When $A$ is Hermitian, $A^{*} = A$ and the identity becomes $\langle A\vec{u}, \vec{v} \rangle = \langle \vec{u}, A\vec{v} \rangle$ . This single equation is the engine behind both theorems on this page: the matrix may be moved across the comma at no cost, and every consequence follows from evaluating one scalar in two ways.

Hermitian matrices have real eigenvalues

Let $A$ be Hermitian with eigenvalue $\lambda$ and eigenvector $\vec{x} \ne \vec{0}$ . Using linearity in the first argument, $\lambda\langle \vec{x}, \vec{x} \rangle = \langle \lambda\vec{x}, \vec{x} \rangle = \langle A\vec{x}, \vec{x} \rangle$ . Applying the self-adjoint shift and then conjugate-linearity in the second argument, the same scalar equals $\langle \vec{x}, A\vec{x} \rangle = \langle \vec{x}, \lambda\vec{x} \rangle = \overline{\lambda}\langle \vec{x}, \vec{x} \rangle$ . Since $\vec{x} \ne \vec{0}$ , positive definiteness of the inner product gives $\langle \vec{x}, \vec{x} \rangle \ne 0$ , so it may be cancelled to leave $\lambda = \overline{\lambda}$ . A complex number equal to its own conjugate has zero imaginary part, so $\lambda \in \mathbb{R}$ .

Hermitian matrices have orthogonal eigenvectors

Now let $\vec{x}$ and $\vec{y}$ be eigenvectors of a Hermitian $A$ for eigenvalues $\lambda$ and $\rho$ with $\lambda \ne \rho$ . The same two-sided evaluation applied to $\langle A\vec{x}, \vec{y} \rangle$ gives $\lambda\langle \vec{x}, \vec{y} \rangle$ on one side and $\overline{\rho}\langle \vec{x}, \vec{y} \rangle$ on the other. The previous theorem has already established that $\rho$ is real, so $\overline{\rho} = \rho$ and the equation reduces to $(\lambda - \rho)\langle \vec{x}, \vec{y} \rangle = 0$ . The eigenvalues are distinct by hypothesis, so the first factor is non-zero and $\langle \vec{x}, \vec{y} \rangle = 0$ . Note the dependency: without the reality theorem the conjugate on $\rho$ could not be removed and the argument would stall.

Why orthogonality is stronger than independence

For a general matrix, eigenvectors from distinct eigenvalues are linearly independent, but they can be arbitrarily close to parallel. Orthogonality is a quantitative strengthening: the eigenvector matrix assembled from an orthonormal set is unitary, so its inverse is its adjoint and its condition number is exactly $1$ . Changing to modal coordinates therefore costs one adjoint multiplication instead of a matrix inversion, introduces no amplification of error, and preserves lengths and angles. For a non-symmetric model the same change of basis can be catastrophically ill-conditioned.

Repeated eigenvalues and the full spectral picture

The orthogonality theorem covers vectors from different eigenvalues; within a single eigenspace of dimension greater than one the basis produced by row reduction is generally not orthogonal. This is not an obstacle: the Gram-Schmidt procedure orthonormalises any basis of a subspace without leaving it, so each eigenspace can be given an orthonormal basis independently. Combined with the fact that a Hermitian matrix is never defective — every eigenvalue satisfies $\gamma_A(\lambda) = \alpha_A(\lambda)$ — this yields an orthonormal basis of $\mathbb{C}^n$ consisting entirely of eigenvectors, and hence a unitary matrix $U$ with $U^{*}AU$ diagonal and real.

What symmetry does not give you

Two limitations are worth stating explicitly. Reality of the eigenvalues says nothing about their sign: a symmetric matrix can have negative eigenvalues, and only positive definiteness — a strictly stronger condition — forces them all positive. And symmetry is not preserved by every operation: the product of two Hermitian matrices is Hermitian only when they commute, so the composition of two self-adjoint models generally loses both guarantees. Generalised eigenvalue problems of the form $K\vec{\phi} = \omega^2 M\vec{\phi}$ with $K$ and $M$ both symmetric retain real eigenvalues and $M$ -orthogonal eigenvectors, but the eigenvectors are not orthogonal in the ordinary inner product.

Building an orthonormal modal basis from a Hermitian matrix

Verify $A = A^{*}$ Check the entrywise condition

[A]_{ij} = \overline{[A]_{ji}}

, including that every diagonal entry is real. Do not assume symmetry that the assembly code does not enforce.

Compute the eigenvaluesAll roots of

p_A(x)

are guaranteed real, so any complex value appearing is a computational artefact and should be traced.

Compute a basis for each eigenspaceReduce

A - \lambda I_n

and read a basis. Every eigenvalue satisfies

\gamma_A(\lambda) = \alpha_A(\lambda)

, so the dimensions will total

n

Orthonormalise within each eigenspaceApply Gram-Schmidt to any eigenspace of dimension greater than one. Vectors from different eigenspaces are already orthogonal and need no treatment.

Assemble the unitary matrix $U$ Place the orthonormal eigenvectors as columns. Then

U^{*}U = I_n

and

U^{*}AU

is real diagonal, with the eigenvalues in the order of the columns.

Equations

Hermitian condition

EQ-EHM-01

A = A^{*} = {\bar{A}}^{t}, [A]_{i j} = \bar{[A]_{j i}}

The defining property. Setting $i = j$ shows every diagonal entry is real; for a real matrix the condition is plain symmetry.

Adjoint shift inside the inner product

EQ-EHM-02

⟨ A \vec{u}, \vec{v} ⟩ = ⟨ \vec{u}, A^{*} \vec{v} ⟩, A = A^{*} \Rightarrow ⟨ A \vec{u}, \vec{v} ⟩ = ⟨ \vec{u}, A \vec{v} ⟩

Valid for any square matrix on the left; the Hermitian condition collapses it to a statement about $A$ alone.

Reality of the eigenvalues

EQ-EHM-03

λ ⟨ \vec{x}, \vec{x} ⟩ = ⟨ A \vec{x}, \vec{x} ⟩ = ⟨ \vec{x}, A \vec{x} ⟩ = \bar{λ} ⟨ \vec{x}, \vec{x} ⟩ \Rightarrow λ = \bar{λ} \in ℝ

The cancellation of $\langle \vec{x}, \vec{x} \rangle$ is licensed by positive definiteness, since $\vec{x}$ is an eigenvector and therefore non-zero.

Orthogonality of eigenvectors

EQ-EHM-04

λ \neq ρ, A \vec{x} = λ \vec{x}, A \vec{y} = ρ \vec{y} \Rightarrow (λ - ρ) ⟨ \vec{x}, \vec{y} ⟩ = 0 \Rightarrow ⟨ \vec{x}, \vec{y} ⟩ = 0

The reality theorem is used to replace $\overline{\rho}$ by $\rho$ ; without it the difference of eigenvalues would not factor out.

Unitary diagonalisation

EQ-EHM-05

U^{*} A U = [\begin{matrix} λ_{1} & 0 & \dots & 0 \\ 0 & λ_{2} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & λ_{n} \end{matrix}], U^{*} U = I_{n}, λ_{i} \in ℝ

The columns of $U$ are an orthonormal basis of eigenvectors. Because $U$ is unitary, $U^{-1} = U^{*}$ and the change of basis costs one multiplication.

Rayleigh quotient is real

EQ-EHM-06

R (\vec{x}) = \frac{⟨ A \vec{x}, \vec{x} ⟩}{⟨ \vec{x}, \vec{x} ⟩} \in ℝ, λ_{min} \leq R (\vec{x}) \leq λ_{max}

A direct consequence of self-adjointness. The quotient is stationary exactly at eigenvectors and is the basis of variational eigenvalue estimation.

Modal decoupling

EQ-EHM-07

\vec{x} = \sum_{i}^{=} ⟨ \vec{x}, {\vec{u}}_{i} ⟩ {\vec{u}}_{i}, A \vec{x} = \sum_{i}^{=} λ_{i} ⟨ \vec{x}, {\vec{u}}_{i} ⟩ {\vec{u}}_{i}

Orthonormality makes the expansion coefficients simple inner products, with no linear system to solve. The action of $A$ is then a scalar multiplication in each mode.

Variable Definitions

Symbols used on this page
Symbol	Name	Meaning	Domain / type
$A$	Hermitian matrix	Square matrix satisfying $A = A^{*}$	n x n complex matrix
$A^{*}$	Adjoint	Conjugate transpose of $A$	n x n complex matrix
$λ$	Eigenvalue	Guaranteed real for a Hermitian matrix	real number
$ρ$	Second eigenvalue	A different eigenvalue used in the orthogonality argument	real number
$\vec{x}$	Eigenvector	Non-zero vector with $A\vec{x} = \lambda\vec{x}$	non-zero vector in C^n
$⟨ \vec{u}, \vec{v} ⟩$	Inner product	$\sum_i [\vec{u}]_i \overline{[\vec{v}]_i}$ , linear in the first argument	complex scalar
$U$	Unitary matrix	Matrix whose columns are an orthonormal eigenvector basis	n x n with U*U = I
$R (\vec{x})$	Rayleigh quotient	Real-valued ratio bounded by the extreme eigenvalues	real number
$\bar{z}$	Complex conjugate	Sign reversal of the imaginary part of a scalar	complex number

Worked Numerical Example

Problem statement

A three-degree-of-freedom lumped model produces the symmetric stiffness matrix $K$ . Confirm that its eigenvalues are real, that its mode shapes are mutually orthogonal, and assemble the orthogonal matrix that diagonalises it.

Confirm the Hermitian condition
All entries are real, so the adjoint reduces to the transpose. Comparing $[K]_{ij}$ with $[K]_{ji}$ across the three off-diagonal pairs confirms $K = K^{t}$ , hence $K = K^{*}$ and both theorems apply.
$K = [\begin{matrix} 7 & - 2 & 0 \\ - 2 & 6 & - 2 \\ 0 & - 2 & 5 \end{matrix}] = K^{t}$
Compute the spectrum
Expanding along the first row and factoring gives a polynomial with three real roots — as the reality theorem guarantees in advance, so a complex root here would have signalled an arithmetic error rather than a mathematical discovery.
$p_{K} (x) = - x^{3} + 18 x^{2} - 99 x + 162 = - (x - 3) (x - 6) (x - 9)$
Compute one eigenvector per eigenvalue
Each eigenvalue is simple, so each eigenspace is a line. Reducing $K - \lambda I_3$ for each root and clearing fractions gives integer representatives:
${\vec{v}}_{1} = [\begin{matrix} 1 \\ 2 \\ 2 \end{matrix}] (λ = 3), {\vec{v}}_{2} = [\begin{matrix} - 2 \\ - 1 \\ 2 \end{matrix}] (λ = 6), {\vec{v}}_{3} = [\begin{matrix} 2 \\ - 2 \\ 1 \end{matrix}] (λ = 9)$
Verify orthogonality directly
The entries are real, so each inner product is an ordinary dot product. All three pairings vanish, exactly as the theorem requires for distinct eigenvalues of a symmetric matrix.
$⟨ {\vec{v}}_{1}, {\vec{v}}_{2} ⟩ = - 2 - 2 + 4 = 0, ⟨ {\vec{v}}_{1}, {\vec{v}}_{3} ⟩ = 2 - 4 + 2 = 0, ⟨ {\vec{v}}_{2}, {\vec{v}}_{3} ⟩ = - 4 + 2 + 2 = 0$
Normalise to unit length
Each vector happens to have squared norm $1 + 4 + 4 = 9$ , so dividing by $3$ produces a unit vector in every case. Orthogonality is unaffected by scaling, so the set is now orthonormal.
${\vec{u}}_{i} = \frac{1}{3} {\vec{v}}_{i}, ⟨ {\vec{u}}_{i}, {\vec{u}}_{j} ⟩ = {\begin{cases} 1 & i = j \\ 0 & i \neq j \end{cases}$
Assemble the orthogonal matrix
Placing the normalised eigenvectors as columns gives a matrix whose transpose is its inverse. Confirming $Q^{t}Q = I_3$ requires exactly the nine inner products already computed.
$Q = \frac{1}{3} [\begin{matrix} 1 & - 2 & 2 \\ 2 & - 1 & - 2 \\ 2 & 2 & 1 \end{matrix}], Q^{t} Q = I_{3}$
Diagonalise
Because $Q^{-1} = Q^{t}$ , the similarity transformation requires no inversion at all — a transpose and two matrix products. The result is diagonal with the eigenvalues in column order.
$Q^{t} K Q = [\begin{matrix} 3 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 9 \end{matrix}]$
Check a Rayleigh quotient
Take the trial vector $\vec{w} = (1,0,0)$ . Then $\langle K\vec{w}, \vec{w} \rangle = 7$ and $\langle \vec{w}, \vec{w} \rangle = 1$ , so $R(\vec{w}) = 7$ . As required, this lies between $\lambda_{\min} = 3$ and $\lambda_{\max} = 9$ , and it is real for every trial vector because the matrix is self-adjoint.

Result

The model has three real natural stiffnesses $3$ , $6$ and $9$ and three mutually orthogonal mode shapes. The transformation into modal coordinates is an orthogonal matrix, so it preserves energy, requires no inversion and introduces no numerical amplification. In modal coordinates the three degrees of freedom are completely decoupled: each responds independently with its own stiffness, which is why symmetric formulations are preferred wherever the physics permits them.

Applications & Industry Use

Structural engineering

Real natural frequencies and orthogonal modes

Stiffness and mass matrices assembled from a variational principle are symmetric by construction, so natural frequencies come out real and mode shapes come out mutually orthogonal with respect to the mass matrix. Modal superposition relies on that orthogonality: it is what allows a multi-degree-of-freedom response to be assembled from independent single-degree-of-freedom solutions.

Computing & data

Principal component analysis

A covariance matrix is symmetric and positive semi-definite. Reality of the eigenvalues means the variance along each principal direction is a genuine number rather than a complex artefact, and orthogonality of the eigenvectors means the principal components are uncorrelated. Both properties are assumed silently by every implementation of the method.

Quantum and photonic engineering

Observables and measurement outcomes

Physical observables are represented by Hermitian operators precisely because measured values must be real. The reality theorem is the mathematical statement of that requirement, and the orthogonality theorem gives the mutually exclusive measurement outcomes associated with distinct eigenvalues.

Signal processing

Autocorrelation matrices and subspace methods

The autocorrelation matrix of a wide-sense stationary process is Hermitian and Toeplitz. Subspace direction-finding methods split its orthonormal eigenvector basis into signal and noise subspaces, a separation that only makes sense because the eigenvectors are guaranteed orthogonal and the eigenvalues real and orderable.

Mechanical engineering

Principal stresses and strains

The Cauchy stress tensor is symmetric as a consequence of moment equilibrium. Its eigenvalues are therefore real principal stresses and its eigenvectors mutually perpendicular principal directions, on which shear vanishes. Failure criteria are stated in terms of these real values, and the perpendicularity of the principal axes is what makes Mohr's circle constructions valid.

Electrical engineering

Passive network matrices

Impedance and admittance matrices of reciprocal passive networks are symmetric, and their Hermitian parts govern power dissipation. Real eigenvalues of the Hermitian part correspond to real dissipated powers, and the orthogonality of the associated modes underlies the decomposition of a multi-port network into independent power channels.

Design Considerations

Enforce symmetry in assembly, do not merely assume it

Finite element assembly, numerical differentiation and empirical covariance estimation all produce matrices that should be symmetric but frequently are not, to within rounding or worse. Symmetrise explicitly by replacing $A$ with $\tfrac{1}{2}(A + A^{*})$ , or assemble only one triangle and mirror it. Every guarantee on this page evaporates when the condition fails by more than roundoff.

Use the symmetric eigensolver, not the general one

Routines such as numpy.linalg.eigh and LAPACK's xSYEV and xHEEV read only one triangle, return real eigenvalues in ascending order and produce an exactly orthonormal eigenvector matrix. They are roughly four times faster than the general solver and substantially more accurate. Passing a symmetric matrix to a general eigensolver wastes both.

Orthonormalise within repeated eigenspaces

The orthogonality theorem covers only vectors from different eigenvalues. Where an eigenvalue is repeated, the basis returned by row reduction will generally not be orthogonal, and it must be passed through Gram-Schmidt before the eigenvector matrix is assembled. Numerical symmetric eigensolvers do this internally and always return an orthonormal set.

Do not confuse real with positive

Reality of the eigenvalues is a consequence of symmetry alone; positivity requires positive definiteness, which is a strictly stronger condition. A symmetric matrix with a negative eigenvalue is entirely ordinary, and in a stability analysis it is the signature of an unstable equilibrium. Where positivity is needed, test it with a Cholesky factorisation rather than inferring it from symmetry.

Handle the generalised problem correctly

Vibration problems take the form $K\vec{\phi} = \omega^2 M\vec{\phi}$ with $K$ and $M$ symmetric and $M$ positive definite. The eigenvalues remain real, but the eigenvectors are orthogonal with respect to the $M$ -inner product, not the standard one. Reduce to a standard problem with a Cholesky factor of $M$ , or use a dedicated generalised symmetric solver; do not test ordinary orthogonality and conclude the computation is wrong.

Exploit the perfect conditioning

For a Hermitian matrix, an eigenvalue perturbation is bounded by the norm of the matrix perturbation — the eigenvalues are perfectly conditioned. This is a strong guarantee that no non-normal matrix enjoys, and it justifies quoting computed natural frequencies of a symmetric model to near machine precision, whereas the same claim for a non-symmetric model would require an explicit condition estimate.

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 for the complex conjugate bar, the adjoint superscript and the angle-bracket inner product, all of which carry the argument on this page.
`LAPACK reference implementation`	Linear Algebra PACKage symmetric and Hermitian drivers	`xSYEV`, `xHEEV` and the divide-and-conquer variants exploit self-adjointness to return real ordered eigenvalues and an orthonormal eigenvector matrix at roughly a quarter of the general cost.
`ISO 7626`	Mechanical vibration and shock — Experimental determination of mechanical mobility	Frequency response measurements are interpreted through modal superposition, which requires the mode shapes of the symmetric structural model to be mutually orthogonal.
`IEEE 754-2019`	IEEE Standard for Floating-Point Arithmetic	Determines the residual asymmetry introduced by finite-precision assembly, which must be smaller than the tolerance at which a matrix is accepted as Hermitian.
`ISO/IEC 40314`	Mathematical Markup Language (MathML) Version 3.0	Encodes the conjugate accents, adjoint superscripts and bracket notation as semantic markup so that the self-adjointness condition is machine-readable.

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
Real symmetric binary64, packed or lower-triangle storage	Stiffness, mass, covariance and Gram matrices, which are real and symmetric by construction.	Halves storage and enforces symmetry structurally, but requires the calling code to respect the triangle convention or results will silently reflect the wrong half.
Complex Hermitian binary64	Frequency-domain models, quantum observables and autocorrelation matrices of complex baseband signals.	Doubles storage relative to the real case and requires that diagonal entries be forced exactly real, since a small imaginary part there breaks the Hermitian property.
Banded symmetric storage	One-dimensional chains, beams and finite difference operators where coupling is local.	Reduces both storage and eigensolver cost to near linear in $n$ , but only for genuinely narrow bandwidth; a single long-range coupling destroys the advantage.
Exact rational symmetric	Small models where eigenvalues must be certified real and orthogonality demonstrated exactly, as in this page's example.	Gives unambiguous results, but the eigenvalues of a symmetric integer matrix are usually irrational, so exactness ends at the characteristic polynomial.
Sparse symmetric with Lanczos iteration	Very large symmetric systems where only the extreme eigenvalues and their modes are required.	Extremely efficient for a few eigenpairs and preserves symmetry throughout, but loses orthogonality among computed vectors without explicit reorthogonalisation.
Cholesky-reduced generalised form	Generalised problems $K\vec{\phi} = \omega^2 M\vec{\phi}$ with $M$ symmetric positive definite.	Converts to a standard symmetric problem preserving real eigenvalues, at the cost of one factorisation and a loss of sparsity in the transformed matrix.

Manufacturing Notes

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

Testing the Hermitian property numerically

Compare $\lVert A - A^{*} \rVert$ against a tolerance scaled by $\lVert A \rVert$ and the machine epsilon, rather than testing entries for exact equality. Residual asymmetry at the level of a few units in the last place is the expected outcome of any floating-point assembly; asymmetry orders of magnitude larger indicates a modelling or indexing error that symmetrisation would mask rather than fix.

Cost of the symmetric eigensolver

A Hermitian matrix is reduced to real tridiagonal form by unitary similarity in about $\tfrac{4}{3}n^3$ operations, after which the tridiagonal eigenproblem is solved by QR or divide-and-conquer for a small additional cost. Eigenvalues alone come at roughly $\tfrac{4}{3}n^3$ ; adding eigenvectors brings the total to around $9n^3$ , still well below the $25n^3$ typical of the general case.

Library behaviour

numpy.linalg.eigh and scipy.linalg.eigh read only the lower triangle by default, return eigenvalues as a real array in ascending order, and guarantee an orthonormal eigenvector matrix. scipy.linalg.eigh also accepts a second matrix argument to solve the generalised symmetric problem directly. Using numpy.linalg.eig on a symmetric matrix returns complex-typed eigenvalues with small imaginary parts and a non-orthonormal eigenvector matrix, and should be avoided.

Verifying an orthonormal modal basis

Three checks close the loop. Confirm $\lVert U^{*}U - I_n \rVert$ is at machine level. Confirm the residual $\lVert AU - U\Lambda \rVert$ is small relative to $\lVert A \rVert$ . Confirm the eigenvalues sum to the trace. Together these validate orthonormality, correctness of the eigenpairs and completeness of the spectrum.

Orthonormalisation inside a repeated eigenspace

Where an eigenvalue is repeated, apply the Gram-Schmidt procedure to the eigenspace basis, preferably in its modified form which subtracts each projection as it is computed rather than all at once. The classical form loses orthogonality rapidly in floating point. A QR factorisation of the eigenspace basis matrix achieves the same result more stably and is the preferred implementation.

Failure Modes & Common Mistakes

Failure modes, root causes and prevention
Failure mode / mistake	Impact	Root cause	Prevention & detection
Assuming symmetry that assembly does not deliver	high	Numerically differentiated Jacobians, empirical covariances and mixed-element assemblies that are only approximately symmetric.	Measure $\lVert A - A^{*}\rVert / \lVert A \rVert$ and symmetrise explicitly, or fix the assembly. Every guarantee on this page requires the exact condition.
Expecting orthogonality within a repeated eigenspace	medium	Applying the orthogonality theorem to two basis vectors of the same eigenspace, where it does not apply.	Orthonormalise each repeated eigenspace with Gram-Schmidt or a QR factorisation. Only vectors from different eigenvalues are automatically orthogonal.
Reading real eigenvalues as positive eigenvalues	high	Conflating symmetry with positive definiteness, then treating a negative eigenvalue as an error.	Symmetry gives reality only. Test definiteness separately with a Cholesky attempt or by inspecting the sign of the smallest eigenvalue.
Ignoring the $M$ -inner product in a generalised problem	high	Testing ordinary orthogonality of mode shapes from $K\vec{\phi} = \omega^2 M\vec{\phi}$ and concluding the solver is faulty.	Test $\vec{\phi}_i^{*}M\vec{\phi}_j = 0$ instead. Mode shapes are $M$ -orthogonal, not orthogonal in the standard inner product, unless $M$ is a multiple of the identity.
Complex diagonal entries in a Hermitian matrix	medium	Constructing a Hermitian matrix programmatically and leaving a small imaginary part on the diagonal.	Force the diagonal exactly real when building the matrix. Hermitian solvers ignore the imaginary part of the diagonal, so the error is silent and the returned decomposition will not reproduce the input matrix.
Using a general eigensolver on a symmetric matrix	medium	Calling a one-size-fits-all routine without checking the structure of the matrix.	Dispatch on the structure. The symmetric solver is faster, more accurate, and returns exactly the real ordered eigenvalues and orthonormal vectors the theory promises.
Assuming a product of Hermitian matrices is Hermitian	medium	Composing two self-adjoint models and expecting real eigenvalues from the composition.	$(AB)^{} = B^{}A^{*} = BA$ , which equals $A B$ only when the matrices commute. Symmetrise the product or reformulate the problem as a generalised eigenvalue problem.
Losing orthogonality in an iterative solver	low	Lanczos iteration without reorthogonalisation, where computed vectors drift out of orthogonality and spurious duplicate eigenvalues appear.	Apply full or selective reorthogonalisation, and treat repeated eigenvalues returned by a Lanczos run as suspect until confirmed by an independent method.

FAQs

Why does a Hermitian matrix have real eigenvalues when a general complex matrix does not?

Because the same scalar $\langle A\vec{x}, \vec{x} \rangle$ can be evaluated in two ways. Pulling $\lambda$ out of the first argument gives $\lambda\langle \vec{x}, \vec{x} \rangle$ ; using self-adjointness to move $A$ into the second argument and pulling $\lambda$ out from there gives $\overline{\lambda}\langle \vec{x}, \vec{x} \rangle$ . Cancelling the non-zero factor forces $\lambda = \overline{\lambda}$ , which is exactly the statement that $\lambda$ is real.

Is a real symmetric matrix the same thing as a Hermitian matrix?

A real symmetric matrix is a Hermitian matrix, since conjugation does nothing to real entries and the adjoint reduces to the transpose. The converse is not true: a Hermitian matrix may have complex off-diagonal entries, related by conjugation across the diagonal. Both theorems on this page apply equally to either case.

Are the eigenvectors of a Hermitian matrix always orthogonal?

Eigenvectors belonging to different eigenvalues always are. Within a single eigenspace of dimension greater than one, an arbitrary basis need not be orthogonal, but it can always be orthonormalised by Gram-Schmidt without leaving the eigenspace. Assembling the orthonormalised bases of all eigenspaces gives an orthonormal basis of the whole space.

Can a Hermitian matrix be defective?

No. Every eigenvalue of a Hermitian matrix satisfies $\gamma_A(\lambda) = \alpha_A(\lambda)$ , so there is always a full set of $n$ independent eigenvectors and the matrix is always diagonalisable — in fact unitarily diagonalisable. This is why symmetric formulations remove an entire class of numerical and modelling difficulty.

Does reality of the eigenvalues mean they are all positive?

No. Symmetry alone gives reality; positivity requires positive definiteness, which asks that $\langle A\vec{x}, \vec{x} \rangle > 0$ for every non-zero $\vec{x}$ . A symmetric matrix with mixed signs in its spectrum is perfectly normal and, in an equilibrium analysis, is the signature of a saddle point.

Why are the mode shapes from a vibration problem not orthogonal in the usual sense?

Because the generalised problem $K\vec{\phi} = \omega^2 M \vec{\phi}$ is not a standard eigenvalue problem. Its eigenvectors are orthogonal with respect to the inner product defined by the mass matrix, satisfying $\vec{\phi}_i^{*}M\vec{\phi}_j = 0$ for $i \ne j$ . Ordinary orthogonality is recovered only when $M$ is a multiple of the identity.

What practical advantage does an orthonormal eigenvector basis give?

The eigenvector matrix $U$ becomes unitary, so $U^{-1} = U^{*}$ and the change into modal coordinates requires a transpose rather than an inversion. Its condition number is exactly one, so no error is amplified by the transformation, and expansion coefficients are simple inner products rather than the solution of a linear system.

References

Beezer, R. A. A First Course in Linear Algebra, Version 0.70. University of Puget Sound, 2006. Section PEE, Subsection EHM. Licensed under the GNU Free Documentation License v1.2.
Horn, R. A. and Johnson, C. R. Matrix Analysis, 2nd edition. Cambridge University Press, 2012. Chapter 4, Hermitian Matrices and Symmetric Matrices.
Parlett, B. N. The Symmetric Eigenvalue Problem. Society for Industrial and Applied Mathematics, 1998.
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

Show a $2 \times 2$ Hermitian matrix with complex off-diagonal entries, compute its real eigenvalues, and verify that its eigenvectors are orthogonal under the complex inner product.
Prove that the eigenvalues of a skew-Hermitian matrix are purely imaginary, using the same two-sided inner product argument.
Why are the eigenvalues of a Hermitian matrix perfectly conditioned, and what does the Bauer-Fike bound say for a non-normal matrix?
Derive the $M$ -orthogonality of mode shapes in the generalised problem $K\vec{\phi} = \omega^2 M\vec{\phi}$ and show how to reduce it to a standard symmetric problem.
Explain the variational characterisation of eigenvalues through the Rayleigh quotient and how it gives the Courant-Fischer min-max theorem.
What is the cheapest reliable numerical test that a large sparse matrix is Hermitian to within roundoff?

Related Calculators

Hermitian Property Checker

Test $A = A^{*}$ entrywise with a scaled tolerance and report the residual asymmetry.

Symmetric & Hermitian Eigensolver

Return real ordered eigenvalues and an orthonormal eigenvector matrix for a self-adjoint input.

Rayleigh Quotient Calculator

Evaluate $R(\vec{x})$ for a trial vector and bracket it against the extreme eigenvalues.

Gram-Schmidt Orthonormaliser

Orthonormalise a basis of a repeated eigenspace using the modified algorithm.