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Approximation of linear operators in the 2norm \Lambda A.C. Antoulas

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
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


Source: Antoulas, Athanasios C. - Department of Electrical and Computer Engineering, Rice University


Collections: Engineering