Hadamard functions that preserve inverse \(M\)-matrices (Q2910967)

Let \(n\geq 2\) be an integer. A real \(n\)--by-\(n\) matrix \(A\in {\mathcal M}_n({\mathbb R})\) is an \(M\)-matrix if it is of the form \((\alpha I -P)\) for some entry-wise nonnegative matrix \(P\) and some \(\alpha>\rho(P)\), the spectral radius of \(P\). Let \({\mathcal M}^{-1}\), \({\mathcal P}\), and \(bi{\mathcal P}\subseteq {\mathcal M}_n({\mathbb R})\) denote the set of inverse \(M\)-matrices, the set of potential matrices (inverses of row diagonally dominant \(M\)-matrices) and the set of bipotential matrices (inverses of row and column diagonally dominant \(M\)-matrices).NEWLINENEWLINEThe authors study the conditions which force the above mentioned classes to be invariant under matrix Hadamard functions \(f:\bigl(a_{ij} \bigr)_{ij}\mapsto \bigl(f(a_{ij}) \bigr)_{ij}\), induced by a given real function \(f:[0,\infty)\to[0,\infty)\).NEWLINENEWLINEPositive Hadamard powers \(f_\alpha(x)=x^\alpha\), \(\alpha>0\), are considered first. The authors show that if the Hadamard logarithm of a matrix \(W\), that is, \(V=\ln W\) is bipotential then all positive Hadamard powers \(W^{(a)}=f_\alpha(W)=\text{exp} (\alpha V)\) of \(W\) are bipotential, and in particular \(V\in bi{\mathcal P}\) implies \(\text{exp} V\in bi{\mathcal P}\). This is no longer true for potential matrices as shown by an example. However, if \(V=\ln W\) belongs to \({\mathcal P}\) and simultaneously to a certain limit class \({\mathcal P}_\sigma\) of potential matrices then all positive powers of \(W\) again remain inside \({\mathcal P}\). In case of inverse \(M\)-matrices, similar conclusions are derived except one has to additionally assume that at least one positive power of \(W\) is invertible. This extends the results obtained by \textit{S. Chen} [Linear Algebra Appl. 422, No. 2--3, 477--481 (2007; Zbl 1117.15021)] where it was shown that \({\mathcal M}^{-1}\) is closed under Hadamard real powers greater than one.NEWLINENEWLINEThe stability of the classes \({\mathcal C}\in\{{\mathcal M}^{-1},{\mathcal P},bi{\mathcal P}\}\), as well as the stability of sub-Markov potential and sub-Markov bipotential matrices under general Hadamard functions are also investigated. The authors show that a Hadamard function \(f\) leaves invariant a class \({\mathcal C}\in\{{\mathcal M}^{-1},{\mathcal P},bi{\mathcal P}\}\) if and only if (i) \(\det f(A)>0\) for each \({\mathcal C}\)-matrix \(A\) and (ii) \(\det f(A)\geq 0\) for each entry-wise nonnegative \(A\) obtained from \((n+1)\)--by--\((n+1)\) \({\mathcal C}\)-matrix \(\hat{A}\) by deleting its first column and second row. Alternatively, in the case \(V\in bi{\mathcal P}\), the sufficient and necessary conditions that \(f(V)\in bi{\mathcal P}\) for a given Hadamard function \(f\) are also stated in terms of potential vectors of mass at most one, that is, vectors \(v=Vz\) where \(z\) is entry-wise nonnegative with sum of its entries at most one.