Pinchings and Norms of Scaled Triangular Matrices 1 Pinchings and Norms of Scaled Triangular Matrices

Suppose U is an upper-triangular matrix, and D a nonsingular diagonal matrix whose diagonal entries appear in nondescending order of magnitude down the diagonal. It is proved that kD UDk kUk for any matrix norm that is reduced by a pinching. In addition to known examples { weakly unitarily invariant norms { we show that any matrix norm de ned by kAk def = max x 6=0; y 6=0 Re (x Ay) (x) (y) ; where ( ) and ( ) are two absolute vector norms, has this property. This includes `p operator norms as a special case. This report is available on the web at http://www.ms.uky.edu/~rcli/. 2Indian Statistical Institute, New Delhi 110 016, India. email: [email protected] 3Computer Science Division and Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720. email: [email protected] 4Department of Mathematics, University of Kentucky, Lexington, KY 40506 ([email protected]). This work was supported in part by the National Science Foundation under Grant No. ACI-9721388 and by the National Science Foundation CAREER award under Grant No. CCR-9875201. Pinchings and Norms of Scaled Triangular Matrices Rajendra Bhatia William Kahan y Ren-Cang Li z May 2000