My favorite piece of numerical linear algebra is the pseudospectrum. In some sense, this is a set that gives you an indication of whether your eigenvalue compu

My favorite piece of numerical linear algebra is the pseudospectrum. In some sense, this is a set that gives you an indication of whether your eigenvalue computations will be accurate or not. Understanding non-normal operators really requires an understanding of this object. It also has a surprising connection to the local solvability of pdes and pseudodifferential operators by helping to construct quasimodes ("almost solutions").