Pinchings and norms of scaled triangular matrices

Bhatia, Rajendra ; Kahan, William ; Li, Ren-Cang (2002) Pinchings and norms of scaled triangular matrices Linear and Multilinear Algebra, 50 (1). pp. 15-21. ISSN 0308-1087

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Suppose U is an upper-triangular matrix, and D a nonsingular diagonal matrix whose diagonal entries appear in non- descending order of magnitude down the diagonal. It is proved that ||D-1UD||≥||U|| 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 defined by ||A||=def/max RE(x* Ay)/x≠o, y≠oΦ(x)φ(y) where Φ(·) and ψ(·) are two absolute vector norms, has this property. This includes ellp operator norms as a special case.

Item Type:Article
Source:Copyright of this article belongs to Taylor and Francis Ltd.
Keywords:Triangular Matrix; Scaling; ℓ p Operator Norm; Unitarily Invariant Norm; Pinching Inequality
ID Code:97485
Deposited On:20 Feb 2013 09:02
Last Modified:20 Feb 2013 09:02

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