 
Summary: Approximation of linear operators in the 2norm \Lambda
A.C. Antoulas
Department of Electrical and Computer Engineering
Rice University
Houston, Texas 772511892, USA
email: aca@rice.edu fax: +17135245237
September 1, 1997
Abstract
The problems of approximating in the 2induced norm, linear oparators which are (1)
finitedimensional, unstructured, and (2) infinitedimensional structured (Hankel), have been
solved. The solutions of these two problems exhibit striking similarities. These similarities
suggest the search of a unifying framework for the approximation of linear operators in the
2induced norm.
\Lambda Note submitted to LAA for the Challenges in Matrix Theory Issue.
1 Finitedimensional operators
Given a matrix A 2 R n\Thetan , there exist unitary matrices U; V 2 R n\Thetan , such that
A = U \SigmaV \Lambda where \Sigma = diag (oe 1 ; \Delta \Delta \Delta ; oe n ) 2 R n\Thetan ; oe 1 (A) – \Delta \Delta \Delta – oe n (A) – 0 (1.1)
This is the singular value decomposition (SVD) of the matrix A; the oe i 's are the singular values
of A; oe 1 (A) is the 2induced norm of A. The columns of U , V
