Pinchings and Norms of Scaled Triangular Matrices |
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Authors: | Rajendra Bhatia William Kahan Ren-Cang Li |
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Institution: | 1. Indian Statistical Institute , New Delhi, 110 016, India;2. Computer Science Division and Department of Mathematics , University of California at Berkeley , Berkeley, CA, 94720, USA;3. Department of Mathematics , University of Kentucky , Lexington, KY, 40506, USA |
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Abstract: | 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 $$\|D^{-1}UD\|\ge\|U\|$$ 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 $$\| A \|^{\underline{\underline {{\rm def}}} } \mathop {\max }\limits_{x \ne 0,y \ne 0} {{{\mathop{\rm Re}\nolimits} (x^*Ay)} \over {\phi (x)\psi (y)}},$$ where θ (.) and y (.) are two absolute vector norms, has this property. This includes l p operator norms as a special case. |
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Keywords: | Triangular Matrix Scaling ? p Operator Norm Unitarily Invariant Norm Pinching Inequality |
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