Index of Hadamard multiplication by positive matrices
The author introduces two notions of index of a positive complex \(n\times n\) matrix \(A\) in terms of its action on the positive matrices by Hadamard multiplication: \[ \begin{aligned} I_A& =\max\{\lambda \geq 0: \|A\circ B\|\geq \lambda\|B\|, \;0 \geq B \in M_n\}, \\ II_A& =\max\{\lambda \geq 0: A\circ B \geq \lambda B, \;0 \geq B \in M_n\}.\end{aligned} \] He gives formulae to compute these indices. Originally the numbers \(I_A\) and \(II_A\), for positive matrices were studied in order to get sharp constants for some operator inequalities [see for example \textit{L. Livshits} and \textit{S. C. Ong}, ibid. 183, 117-129 (1993; Zbl 0809.47019)]. Let \(M_n\) be the \({\mathbb C}^{*}\)-algebra of \(n\times n\) complex matrices over \({\mathbb C}\), \(P(n) = \{A \in M_n: A \geq 0\}\) the set of positive (or non-negative) matrices and the map \(\Phi_{A}: M_n \to M_n\) is given by \(\Phi_{A}(B) = A\circ B\) for all \(B \in M_n\). On studying the formulae for \(I_A\) and \(II_A\) the author finds that they are given in terms of positive matrix completions and factorizations of \(A\) (see Theorem 3.11), very similar as those formulae for the norm of \(\Phi_{A}\). Furthermore, his formulae are closely related with the harmonic mean \(M:({\mathbb R}^{+})^{n} \to {\mathbb R}^{+}\) given by \[ M(d_1, \ldots d_n) = \left(\sum_{i=1}^n d_i^{-1}\right)^{-1}, \quad (d_1, \ldots d_n) \in ({\mathbb R}^{+})^{n}. \] Actually he uses its mean properties in order to compute both indexes for some examples of positive matrices (namely, diagonal or block diagonal matrices) and also to characterize those matrices \(A \in P(n)\) such that \(I_A = 0\). Moreover a complete description is given in the \(2 \times 2\) case.
scientific article; zbMATH DE number 1331368
scientific article