Existence, Uniqueness and Algorithms for Matrix Unitary Reduction to Semiseparable Form

Bevilacqua, Roberto and Del Corso, Gianna M. (2003) Existence, Uniqueness and Algorithms for Matrix Unitary Reduction to Semiseparable Form. Technical Report del Dipartimento di Informatica . Università di Pisa, Pisa, IT.

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Abstract

This paper concerns the study of a unitary transformation from a generic 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 proof of the existence of such a transformation and then we discuss the uniqueness of such transformation proving an Implicit-$Q$ Theorem for semiseparable matrices. Finally, we study structural properties of the factors of the $QR$-decomposition of a semiseparable matrix. These properties allows us to design a method based on the $QR$ 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 has been accomplished, if one is interested in computing the eigenvalues each further $QR$ iteration can be done in linear time.

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