Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Resolvent — where it appears

The inverse of zI − A as a function of z, whose norm is one over the distance to the spectrum exactly when the matrix is normal. Its norm is what a pseudospectrum is a level set of, and its distance from one over the gap to the spectrum is a measure of non-normality.

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

Also named here as transient growth — the same set of essays touches all of them, so they are one junction rather than several.

darker is smaller: 10⁻¹⁰, 10⁻⁶, 10⁻³, 10⁻¹one spectrum, two matricesspectral radius0.8reach of the 10⁻³ level1.3eigenvalues, all at0.8the circle is |z| = 1and every eigenvalue is well inside it

The eigenvalues that are not there

For a normal matrix the resolvent norm is exactly one over the distance to the nearest eigenvalue, so a picture of it carries nothing the spectrum did not. Move one entry above the diagonal and the region a perturbation of 10⁻⁸ can put an eigenvalue into stops being a disc and reaches out past the unit circle, while every eigenvalue stays at 0.8.

spectra · Non-normality 027548110813510⁻⁶10⁻⁴10⁻²110²10⁴10⁶power‖Aᵏ‖Kreiss constant 6760e · n · K‖Aᵏ‖ρᵏtwo routes to one peakspectral radius0.8peak of ‖Aᵏ‖2·10⁴Kreiss constant6757e · n · K1.1·10⁵everything here decays in the endand one of these curves says how much first

A spectral radius that grows first

ρ(A) below one guarantees that the powers of A go to zero and says nothing about what they do on the way. Here they rise by a factor of twenty thousand before turning over, and the peak is bracketed above and below by a constant computed from the resolvent norms outside the unit circle — two routes to one number, one through the plane and one through the powers.

spectra · Non-normality

Named alongside it

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

Non-normalityPseudospectrumTransient growthAsymptotic analysisConvergence rateEigenvalue condition numberThe Kreiss constantNon-normal matricesPerturbationThe real Schur formSingular valuesSpectral radius

All concepts