Details

Knowledge of the spectrum, the exact location of all or some of the eigenvalues, whether an eigenvalue is real, in the left half-plane, or not in a specified region, or anything else like that is of key importance.

The Russian mathematician Gershgorin became famous for his disks that cover the eigenvalues; they constitute a relatively crude enclosure that compensates by being easy to calculate. However, if all you need is to know if the spectrum is part of a region or not, this crude estimate might already suffice. And if you already have an approximation of the eigensystem, for instance during the course of an iterative procedure like the Jacobi method, the Gershgorin circles of the similarity shrink down until they finally consist of the points of the spectrum alone. This is certainly true only if the matrix is normal, for only then is the Schur similarity actually diagonal. In general the Schur similarity is upper triangular, which means that some of its row or column circles do not have zero diameter, although its diagonal and its spectrum and the spectrum of coincide. Nevertheless, the knowledge of an intermediate stage might already be sufficient.

A recent book of Varga [3] brought these seemingly old ideas back to our attention, providing general extensions of the ideas up to the present. [1] and [2], key books on matrix analysis, give some additional background to the original ideas and their immediate consequences.

Varga's book motivated this Demonstration, but the Demonstration "Gershgorin Circles" by Chris Maes triggered it. Some ideas therefrom were the starting points for this program, especially the use of RegionPlot with a predicate. That is really a good idea!

What we want to show is as much visual information as possible about the matrix and some easily derivable entities like its hub, its convex hull, the convex hull of its diagonal, its Gershgorin circles, and maybe more, to be visually compared with its spectrum, which a priori is usually not available. The significance or failure of the concepts for a specific matrix should leap to the eyes, especially when using the transition from the matrix to its Schur similarity via a unitarized convex combination of its eigenvectors and the identity , namely, , where the parameter varies between 0 and 1; is the orthogonalization of ; .

Snapshot 0: push buttons one after the other; after any change mouseover the graphics to see explanations

Snapshot 1: typical Gershgorin region based on rows and columns

Snapshot 2: the matrix is not diagonalizable, so the Gershgorin circles show up even for the Schur similarity of , that is,

Snapshot 3: Gershgorin region based on columns, intermediate similarity stage, that is, , spectral and Heinrich circle shown

Snapshot 4: higher dimension, 15, Gershgorin region based on columns, high similarity stage, that is, , Heinrich circle has a large radius, Gershgorin circles still have relatively large radii although all off-diagonal entries seem to be close to zero

This Demonstration does not deal with any of the extensions mentioned in Varga's book. This might be a future topic. Read Varga's book, it is really worthwhile!

[1] R. Zurmühl and S. Falk, Matrizen und ihre Anwendungen, Berlin, Heidelberg: Springer–Verlag, 1984, Teil 1 and Teil 2.

[2] R. A. Horn and C. R. Johnson, Matrix Analysis, New York: Cambridge University Press, 1985.

[3] R. S. Varga, Gershgorin and His Circles, Berlin, Heidelberg: Springer–Verlag, 2004.