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A note on the complex and real operator norms of real matrices. (English) Zbl 0609.15013
This short paper consists of two results, the second a consequence of the first. Let $$a,b\in {\mathbb{R}}^ m$$, and $$u,v\in {\mathbb{R}}^ n$$, and $$1\leq p\leq +\infty$$. Lemma. $$\alpha =\beta$$ where, $\alpha =\sup_{z\in {\mathbb{C}}}\frac{\| u+zv\|_ p}{\| a+zb\|_ p},\text{ and } \beta =\sup_{x\in {\mathbb{R}}}\frac{\| u+xv\|_ p}{\| a+xb\|_ p}.$ Corollary. Let A be an $$m\times n$$ matrix, then $\sup_{\xi \in {\mathbb{R}}^ n-\{0\}}\| A\xi \|_ p/\| \xi \|_ p=\sup_{\zeta \in {\mathbb{C}}^ n-\{0\}}\| A\zeta \|_ p/\| \zeta \|_ p.$
Reviewer: R.W.Shonkwiler
##### MSC:
 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory 15A45 Miscellaneous inequalities involving matrices
##### Keywords:
matrix norm; Lp norm inequality
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