Speaker
Description
Given a symmetric tridiagonal matrix, it has been well--established, that in exact arithmetic, applying some of its eigenvalues as shifts via the $QR$ strategy produces a particular structured matrix where the leading tridiagonal block contains the non--shifted eigenvalues and a trailing diagonal submatrix of the shifted eigenvalues. We will show that the leading tridiagonal block can be created, up to sign differences, by an orthogonal similarity reduction of a certain symmetric arrowhead matrix constructed directly from the non--shifted eigenvalues and associated eigenvectors. This also provides leverage to show a similar result with an upper bidiagonal matrix and a certain triangular arrowhead matrix. Both of the results set the foundation to show, in a unique way, that the well--known implicitly restarted Lanczos method is mathematically equivalent to thick-restarted Lanczos method with a similar extension to singular value problems using the Golub-Kahan Bidiagonalization process.