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

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.