Abstract
This paper concerns the study of a unitary transformation of a generic real symmetric matrix A into a semiseparable matrix. The problem is studied both theoretically and from an algorithmic point of view. In particular, we first give a formal proof of the existence of such a transformation and then we discuss its uniqueness, proving an implicit-Q theorem for semiseparable matrices. Lastly, we study structural properties of the factors of the QR-decomposition of a semiseparable matrix. These properties allow us to design a method based on QR iterations applied to a semiseparable matrix for reducing a symmetric matrix to semiseparable form. This method has the same asymptotic cost of the reduction of a symmetric matrix to tridiagonal form. Once the transformation is accomplished, to compute the eigenvalues each further QR iteration can be done in linear time.
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Bevilacqua, R., Corso, G.M.D. Structural properties of matrix unitary reduction to semiseparable form. Calcolo 41, 177–202 (2004). https://doi.org/10.1007/s10092-004-0093-6
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DOI: https://doi.org/10.1007/s10092-004-0093-6