WOLFRAM|DEMONSTRATIONS PROJECT

Enclosing the Spectrum by Gershgorin-Type Sets

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It is possible to cover the eigenvalues of a general matrix
A
merely by using the entries of
A
to derive some disks that will work in some combination. These disks are called Gershgorin disks after the Russian mathematician.
There are two families of disks; one based on the rows and the other on the columns of
A
. These disks are centered at the diagonal entries of
A
; the radii are the sums of the absolute values of the off-diagonal elements in the corresponding row or column, so usually the disks will have different centers and different radii. The green-yellow filled area is built from unions or intersections of those disks and should cover the spectrum of
A
, that is, the set of eigenvalues of
A
.
You can choose three different types of Gershgorin regions. The first is based on the rows of
A
, the second on the columns, and the third on the intersection of row and column disks. In any case, all corresponding circles are shown and the corresponding Gershgorin region is shown as the green-yellow filled area.
A yellow point with a red boundary marks the mean of the diagonal entries of
A
, which is also the mean of the eigenvalues of
A
. We call this point the hub of
A
, because it is the natural pivot point for all matrices similar to
A
. Indeed, one could claim that the whole question of similarity hinges on this point!
Choosing "Heinrich" shows a blue dashed circle centered at the hub of
A
. Its diameter is derived from the Frobenius norm of
A
, which overestimates the spectral norm of
A
but can be calculated simply from the entries of
A
directly, see [1], Teil 2, p. 237, where it is attributed to Heinrich.
For comparison, by choosing "eigenvalues" to show the hub circle, you can see the smallest circle (dashed red) around the hub of
A
that covers the spectrum.
Moreover, for your convenience, you can see the convex hulls of the entries of
A
(in blue) and of the diagonal entries of
A
(in yellow). The convex hull of the eigenvalues of
A
(in red) will let you judge which (combination of) disks will work.
A family of random complex matrices with dimensions from 2 to 15 is selected to illustrate the ideas. You can choose whether the matrix is diagonalizable or not.
In addition, you can experiment with a unitarily similarized version
A
Sim
of
A
via a parameter κ between 0 and 1, where 0 gives
A
itself and 1 the Schur similarity
A
Schur
of
A
. Values in between show a possible passage from
A
to its Schur similarity, as might occur when using an iterative method like the one named after Jacobi.
Almost every item in the graphics is annotated, so guide the mouse over them to see explanations!