Lanczos Tridiagonalization, Golub-Kahan Bidiagonalization and the Core Problem

Abstract

Consider an orthogonally invariant linear approximation problem Ax ~ b. In 'C.C.

Paige, Z. Strakoš: Core problems in linear algebraic systems (SIAM J.

Matrix Anal. Appl. 27 (2006), pp. 861-875)' it was proved that the partial upper bidiagonalization of the matrix [b,A] determines a core approximation problem that contains the necessary and sufficient information for solving the original problem.

Our contribution derives the fundamental characteristics of the core problem from the known relationship between the Golub-Kahan bidiagonalization, the Lanczos tridiagonalization and the properties of Jacobi matrices.

Keywords

• Lanczos
• Tridiagonalization
• Golub-Kahan
• Bidiagonalization
• Problem