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.

### Structural Properties of Matrix Unitary Reduction to Semiseparable Form

#### Abstract

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.
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Bevilacqua, Roberto; DEL CORSO, GIANNA MARIA
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/11568/186550`
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