Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Davis kahan — where it appears

The theorem bounding how far an invariant subspace moves under a perturbation, by the perturbation divided by the gap to the rest of the spectrum. It is what makes a subspace computable where the individual eigenvectors spanning it are not.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as principal angles, symmetric eigenproblem — the same set of essays touches all of them, so they are one junction rather than several.

10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻⁸10⁻⁵10⁻²gap between the two eigenvalueshow far it movedthe eigenvectorsthe eigenvaluestheir plane‖E‖ / gapone perturbation, three answerseigenvalue shift, spread over the sweep1plane angle, spread over the sweep1eigenvector angle, spread1.6·10⁵the dashed line is Davis–Kahan's ‖E‖/gaptwo of the three never noticed

The gap decides the eigenvector

A symmetric matrix's eigenvalues move by at most the size of the perturbation, whatever the spectrum looks like. Its eigenvectors are governed by a completely different quantity — the distance to the neighbouring eigenvalue — and at a gap of 10⁻⁹ the same perturbation turns them through 27°.

spectra · Eigen conditioning the invariant planesolid: beforedashed: aftersame perturbation, two questionsthe vectors turned, radians0.029the plane turned, radians7.6·10⁻⁸what left the plane5.6·10⁻⁸drawn in the unperturbed plane's own basisa radius is not determined; the circle is

The plane survives what its vectors do not

At a gap of 10⁻⁹ a perturbation of 10⁻⁶ turns the two eigenvectors through half a radian and turns the plane they span through 7.6·10⁻⁸ — a ratio of six million. Ask for the subspace instead of the vectors and a hopeless computation becomes a well-conditioned one, with no change to the arithmetic.

spectra · Invariant subspace

Named alongside it

The objects these essays reach for when they reach for this one.

Eigenvalue gapEigenvector conditioningPerturbationPrincipal anglesSymmetric eigenproblemCondition numberInvariant subspaceOrthogonal projectionWeyl's inequality

All concepts