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Monotonicity of the maximum of inner product norms. (English) Zbl 1100.15013
The monotonicity of the norm \(p(x)=\max (p_{A_1}, \dots , p_{A_m})\) on the field \(\mathbb K^n\) is examined; here \(\mathbb K^n\) is the \(n\)-dimensional real or complex vector space, \(p_{A_i}(x)\) are inner product norms \(p_{A_i}\: x\mapsto (x^*A_ix)^{1/2}\) with positive definite \(A_i\in \mathbb K^{n,n}\) and \(\mathbb K^{n,n}\) is the space of all \(n\times n\) dimensional matrices with entries in \(\mathbb K\).
A theorem on the necessary and sufficient conditions (concerning \(\{ p_{A_1}, \dots , p_{A_m}\}\)) for the monotonicity of \(p(x)\) is proved and the special cases \(\mathbb K=\mathbb R\), \(\mathbb K=\mathbb C\) and \(m=3\) are discussed.
Reviewer: Ivan Saxl (Praha)

MSC:
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
52A21 Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry)
15A63 Quadratic and bilinear forms, inner products
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