Band Generalization of the Golub-Kahan Bidiagonalization, Generalized Jacobi Matrices, and the Core Problem

The concept of the core problem in total least squares (TLS) problems was introduced in [C. C. Paige and Z. Strakoš, SIAM J. Matrix Anal. Appl., 27, 2006, pp. 861–875]. It is based on orthogonal transformations such that the resulting problem decomposes into two independent parts, with one of the parts having trivial (zero) right-hand side and maximal dimensions, and the other part with nonzero right-hand side having minimal dimensions. Extension of this concept to the multiple right-hand sides case AX ≈ B in [I. Hnětynková, M. Plešinger, and Z. Strakoš, SIAM J. Matrix Anal. Appl., 34, 2013, pp. 917–931], which is highly nontrivial, is based on application of the singular value decomposition. In this paper we prove that the band generalization of the Golub– Kahan bidiagonalization proposed in this context by Å. Björck also yields the core problem. We introduce generalized Jacobi matrices and investigate their properties. They prove useful in further analysis of the core problem concept. This paper assumes exact arithmetic.