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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
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