The Gershgorin circle theorem (where “Gershgorin” is sometimes also spelled ” Gersgorin” or “Gerschgorin”) identifies a region in the complex plane that. Does every disk have its own eigenvalue? NO but. Theorem. Let A = [aij ] be an n × n complex matrix. If Di1,Di2,,Dik are Gershgorin discs of A that are. Aug 9, The Gershgorin disc theorem is an elementary result that allows you to make very fast deductions about the locations of eigenvalues.

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Incidentally, since we wrote the paper you linked to above, I noticed an argument for our theorem that bypasses Gershgorin.

The theorem is often stated with a supplementary part that gives further information about the location of the eigenvalues: Unfortunately, the examples one might find on the internet are a bit confusing; What would be the mathematical formula for deriving the eigenvalue estimates?

But I slightly regret it.

It only tells us that each one of the eigenvalues is contained in at least one of the Gerschgorin’s discs. Any square matrix A A whose diagonal entries are big theorrem relative to the rest of the row precisely: I feel ever so slightly resentful about it.

Mon Dec 31 For instance, if b is known to six decimal places and the condition number of A is then we can only be confident that x is accurate to three decimal places. But the theorem says nothing about where those two eigenvalues are within that union.


Example For a diagonal matrixthe Gershgorin discs coincide with the spectrum. Personally I never mention it in undergraduate linear algebra, but I do teach it in a graduate Matrix Analysis course.

Gershgorin circle theorem

ShreevatsaR on August 13, 6: I understand that certain disks are formed, each centered at the diagonal entry, with the radius equal to the summation of absolute values of the associated off-diagonal row entries. ShreevatsaR on August 10, 4: Tom Leinster on August 9, 7: Gershgorin is also an easy consequence of Levy—Desplanques! Did you get taught the Gershgorin disc theorem as an undergraduate?

Monthly 56, Theorems thforem algebra Linear algebra Matrix theory. For very high condition numbers, even very small errors gersngorin to rounding can be magnified to such an extent that the result is meaningless.

Gershgorin Circle Theorem

Tom Leinster on September 16, A proof using complex analysis Argument Principle is clear and mathematically sound. 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 21. ShreevatsaR on August 13, 2: Mark Meckes on August 22, 3: Therefore, applying the triangle inequality.

I think setting some problems about it is a nice theoreem, partly because it makes a valuable point — that you can extract some nontrivial information about invariant quantities eigenvalues associated to a matrix quite directly from the highly non-invariant matrix entries.


Gershgorih instance, it lets you look at the matrix.

Gershgorin Circle Theorem — from Wolfram MathWorld

I definitely never learned the Gershgorin theorem as an undergraduate, or in any class. I would appreciate if someone explains this. Tom Leinster on August 16, The first recurring theme is that a nonsingularity theorem for matrices gives rise to an equivalent eigenvalue inclusion set in the complex plane for matrices, and conversely. I did a lot of reading about matrix analysis when I theodem a grad student and postdoc.

Tom Leinster on August 21, 4: I exaggerate… at least a little! I wrote a blog post about this: Moreover, many posts use MathML, which is, currently only supported in Mozilla. By using this site, you agree to the Terms of Use and Privacy Policy.

Maybe some application in numerical analysis were given, but to be sure I would have to dig up my old notebook. The example from http: Phalguni on July 31, 5: There are related theorems that look at the rows and columns simultaneously.