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The eigenvalues of A must also lie within the Gershgorin discs C j corresponding to the columns of A. 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 ...
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.
Gershgorin's circle theorem itself has a very short proof. A strictly diagonally dominant matrix (or an irreducibly diagonally dominant matrix [2]) is non-singular. A Hermitian diagonally dominant matrix with real non-negative diagonal entries is positive semidefinite. This follows from the eigenvalues being real, and Gershgorin's circle ...
Geroch's splitting theorem (differential geometry) Gershgorin circle theorem (matrix theory) Gibbard–Satterthwaite theorem (voting methods) Girsanov's theorem (stochastic processes) Glaisher's theorem (number theory) Gleason's theorem (Hilbert space) Glivenko's theorem (mathematical logic) Glivenko's theorem (probability) Glivenko–Cantelli ...
By the Gershgorin circle theorem, all of the eigenvalues of a stochastic matrix have absolute values less than or equal to one. Additionally, every right stochastic matrix has an "obvious" column eigenvector associated to the eigenvalue 1: the vector 1 used above, whose coordinates are all equal to 1.
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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.
In numerical linear algebra, the QR algorithm or QR iteration is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a matrix.The QR algorithm was developed in the late 1950s by John G. F. Francis and by Vera N. Kublanovskaya, working independently.