Talk:Gershgorin circle theorem

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shouldn't λ = 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)

[edit] 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)