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Proof. Apply the Theorem to A T while recognizing that the eigenvalues of the transpose are the same as those of the original matrix. Example. For a diagonal matrix, the Gershgorin discs coincide with the spectrum. Conversely, if the Gershgorin discs coincide with the spectrum, the matrix is diagonal.
The Gershgorin circle theorem applies the companion matrix of the polynomial on a basis related to Lagrange interpolation to define discs centered at the interpolation points, each containing a root of the polynomial; see Durand–Kerner method § Root inclusion via Gerschgorin's circles for details.
Pages for logged out editors learn more. Contributions; Talk; Gerschgorin circle theorem
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)
Semyon Aronovich Gershgorin (August 24, 1901 – May 30, 1933) was a Soviet (born in Pruzhany, Belarus, Russian Empire) mathematician.He began as a student at the Petrograd Technological Institute in 1923, became a Professor in 1930, and was given an appointment at the Leningrad Mechanical Engineering Institute in the same year.
English: Gershgorin disk theorem example. This diagram shows the discs in yellow derived for the eigenvalues. The first two disks overlap and their union contains two eigenvalues. The third and fourth disks are disjoint from the others and contain one eigenvalue each.
Steiner used the power of a point for proofs of several statements on circles, for example: Determination of a circle, that intersects four circles by the same angle. [2] Solving the Problem of Apollonius; Construction of the Malfatti circles: [3] For a given triangle determine three circles, which touch each other and two sides of the triangle ...
A common thread in many proofs is the Brouwer fixed point theorem. Another popular method is that of Wielandt (1950). He used the Collatz–Wielandt formula described above to extend and clarify Frobenius's work. [29] Another proof is based on the spectral theory [30] from which part of the arguments are borrowed.
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