Talk:Gershgorin circle theorem

Eigenvalue 0
shouldn't $$\lambda=0$$ also lie in one disc ?
 * Not necessarily, because the discs are centered at the diagonal elements, not at zero. So, the great implication of the theorem is that if the diagonal elements are large enough, i.e. the matrix is suitably diagonally dominant, then the discs do not contain zero, so there is no zero eigenvalue, i.e. the matrix is invertible.
 * Also keep in mind that IF a number lambda is in a disc, it does not mean it's an eigenvalue; rather, if lambda is an eigenvalue, then it's in one of the discs. -- Lavaka (talk) 17:33, 16 November 2007 (UTC)

stronger statement of theorem
I think the theorem has a stronger statement, which is roughly like this:

Take the discs D_i and join all overlapping discs and call these E_j. Then precisely as many eigenvalues lie inside E_j as the number of discs that compose E_j.

Note that this is a weaker claim than saying that each disc D_i contains one eigenvalue.

I'll see if I can find the source of this stronger version.

Also, it should be obvious, but it's probably worth pointing out on the main page, that the radii of the disc, Ri = ∑j ≠ i |aij|, can easily be defined as Ri = ∑j ≠ i |aji| as well (i.e. either row or column sum works).

Lavaka 16:08, 13 November 2007 (UTC)

I'm pretty sure your stronger statement is in Horn and Johnson, Matrix Analysis, Cambridge Univ Press. Unfortunately, my copy is currently in a container. -- Jitse Niesen (talk) 17:29, 13 November 2007 (UTC)

Yeah, I'm pretty sure it's true, and pretty sure it'd be in Horn. I found basically the equivalents in two books: Anne Greenbaum's "Iterative Methods for Solving Systems" as well as Quarteroni, Sacco and Saleri's "Numerical Mathematics", who provide a "Third Gershgorin Thm" as well, which holds for irreducible matrices. I suspect the theorem is also in Franklin's "Matrix Theory" and, perhaps, Golub and Van Loan. Quarteroni et al refer to Atkinson "An Intro to Num. Anal" pp 588 for the proofs. So, if some wants to write it up, I think it definitely belongs on the main page -- I don't have time currently to write it up myself. Lavaka 15:18, 16 November 2007 (UTC)

I think you should also add the stronger version of the theorem, and its proof too if it is not too complicated. Scineram (talk) 12:06, 4 November 2008 (UTC)

Nice talking. —Preceding unsigned comment added by Abelfly (talk • contribs) 06:00, 7 April 2010 (UTC)

If nothing else, the stronger version can be found here: buzzard.ups.edu/courses/2007spring/projects/brakkenthal-paper.pdf — Preceding unsigned comment added by Joelmiller (talk • contribs) 18:01, 23 April 2012 (UTC)

Remark by F. Zhang: Similar proofs like this are seen in many books. However, it's arguable whether such a proof is valid. Two major issues are: (1). What is really meant by "eigenvalue continuity"? (This can be settled by Kato's. Precise meaning is available in Horn and Johnson, Matrix Analysis, 2nd ed, 2013, page 122, Theorem 2.4.9.3.) and (2). The eigenvalue functions, defined with parameter $$t \in [0, 1]$$, are in general neither smooth nor traceable. The algebraic multiplicities of the eigenvalues are indeterminable (or lost) as the eigenvalue functions evolve for $$t \in [0, 1]$$. See Peter Lax, Linear Algebra, 2nd edition, Chapter 9. — Preceding unsigned comment added by Wanttoknow (talk • contribs) 08:08, 13 November 2018 (UTC)

Example
In the example, can't you get a better estimate for the eigenvalues by taking the column sums in the last two cases? It would seem to decrease the radius, at least. —Preceding unsigned comment added by 145.120.14.104 (talk) 14:21, 6 November 2009 (UTC)

request example
"If one of the discs is disjoint from the others then it contains exactly one eigenvalue. If however it meets another disc it is possible that it contains no eigenvalue." -- Is it possible to give an example showing the latter possibility? Zerotalk 07:27, 14 May 2010 (UTC)

The example states that one eigen value appears in each of the discs but those is not necessarily true since the first two discs overlap. So there should be another step that reduces the four discs down to three discs where the first two are unioned. Then state that two eigenvalues values appear in the first disc. I agree with Zero that there should be an additional example that exemplifies the fact that a overlapping disc may contain no eigenvalues values since its eigenvalues value could appear instead inside the otheroverlapping disc. 208.54.85.55 (talk) 21:55, 5 February 2011 (UTC)

query addition
Someone added "Note that we can improve the accuracy of the last two discs by applying the formula to the corresponding columns of the matrix, obtaining D(2,1.2) and D( − 11,2.2)." to the example. Is is valid to take a mixture of row and column disks? I've only ever seen a statement of the theorem for all row disks or all column disks. Zerotalk 00:08, 4 May 2011 (UTC)

The improvement comes from the following: the theorem is true for row disks and column disks. Any part of the complex plane that is not inside of both a row and a column disk cannot have an eigenvalue. So for example, if we have a diagonal matrix and I add one off-diagonal entry, there is one row disk that has nonzero radius, and one column disk that has nonzero radius. If those disks don't overlap at all, then the diagonal entries are the eigenvalues. If they do overlap, then there may be an eigenvalue in the overlap. — Preceding unsigned comment added by Joelmiller (talk • contribs) 17:59, 23 April 2012 (UTC)

Ellipses of Cassini
I recall there being a way to strengthen this using Cassin's ellipses. Is this right? Anybody able to fill in the details? — Preceding unsigned comment added by 146.186.131.40 (talk) 23:06, 17 February 2012 (UTC)

Yes, see this theorem on PlanetMath (Brauer) for the details ! Khromegnome (talk) 18:27, 14 December 2012 (UTC)