Structural Properties of Matrix Unitary Reduction to Semiseparable Form

This paper concerns the study of a unitary transformation from 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 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