## 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
Full Text: