Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Invariant subspace — where it appears

A subspace a matrix maps into itself, which is well defined even where the individual eigenvectors spanning it are not.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

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 1357910⁻⁵10⁻⁴10⁻³10⁻²10⁻¹1eigenvalue λamplificationkeptdiscardeda filter on the starting vectormeasured against the polynomial3.9·10⁻¹¹worst kept direction1best discarded direction0.8the roots are the discarded Ritz valuesand the vertical lines are where they sit

Restarting is a filter

A restart throws away the Ritz values it does not want and begins again from a new starting vector. Written in the eigenbasis, that vector's components have been multiplied by a polynomial with its roots at the discarded values — measured component by component, and agreeing with the polynomial to rounding.

spectra · lanczos 12345610⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹Ritz value, largest firstdistance from 10one vectora block of twoa space, not a ratecopies found, one vector1copies found, block of two2Krylov dimension, one vector16the second copy is not in the spaceat any number of steps

An eigenvalue one vector cannot see

A matrix with an exactly doubled eigenvalue at 10. Twelve Lanczos steps find it once; twenty-four find it once, on a Krylov space of dimension 23 in a 24-dimensional problem. A block of two vectors finds it twice. This is not slow convergence — the second copy is not in the space.

spectra · invariant subspace

Named alongside it

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

Eigenvalue gapEigenvector conditioningKrylov subspaceLanczos algorithmRitz valuesBlock methodsDavis kahanFilter factorsMultiplicityOrthogonal projectionPerturbationPrincipal angles

All concepts