Linear Algebra

Eigenvalues and the Spectral Theorem

When is a matrix guaranteed a full set of eigenvectors? The conditions are less generous than you would hope.

Updated May 2026

An eigenvector is a direction a transformation leaves alone:

A v = λ v

Find enough of them and the transformation becomes trivial. In that basis it is a diagonal matrix, and applying it a thousand times is raising numbers to the thousandth power.

Fig. 1

A symmetric matrix acting on vectors around the circle. Each faint segment joins a vector to its image. Only along the two eigen-directions does a vector come back pointing the same way. Everywhere else it is rotated as well as stretched.

The theorem

Theorem

Spectral theorem

If $A$ is Hermitian ( $A^{†} = A$ ) then:

All eigenvalues are real.
Eigenvectors for distinct eigenvalues are orthogonal.
There is an orthonormal basis of eigenvectors, so $A = U Λ U^{†}$ with $U$ unitary and $Λ$ real diagonal.

Three properties, all following from one symmetry, and between them they are why quantum mechanics uses Hermitian operators for observables: real eigenvalues are measurable numbers, orthogonality makes outcomes distinguishable, and completeness means the basis can express any state.

What diagonalisation is for

Any function of a matrix reduces to the same function of its eigenvalues:

f (A) = U f (Λ) U^{†}

So $e^{A t}$ , the solution operator of every linear ODE system, is computable. The eigenvalues determine the behaviour entirely: negative real parts decay, positive grow, imaginary parts oscillate. Stability analysis is eigenvalue analysis.

Application

Normal modes

Coupled oscillators give $M \ddot{x} = - K x$ . Diagonalising $K$ relative to $M$ decouples the system into independent oscillators, the normal modes. A molecule's vibrational spectrum, a bridge's resonances, and the phonons in a crystal are all this computation.

Degeneracy is a symmetry

Repeated eigenvalues mean the eigenspace has dimension greater than one, and any basis of it will do.

Physically, degeneracy almost always signals a symmetry: hydrogen's $ℓ$ levels are degenerate because the Coulomb potential has a hidden $S O (4)$ symmetry beyond the obvious rotational one. Break the symmetry by applying a field, and the degeneracy lifts, splitting the spectral line. Reading which lines split, and by how much, is how the symmetry gets identified in the first place.

Non-Hermitian matrices misbehave

Without the symmetry, guarantees evaporate. Eigenvalues may be complex, eigenvectors need not be orthogonal, and a matrix may not be diagonalisable at all, and the Jordan form is what remains.

Worse, non-orthogonal eigenvectors make the spectrum a poor guide to behaviour. A matrix with all eigenvalues inside the unit disc, and so asymptotically stable, can still amplify by orders of magnitude before decaying. That transient growth is why some fluid flows go turbulent at Reynolds numbers where linear stability analysis says they should not.