Equilibrants, semipositive matrices, calculation and scaling (Q630507)

A real matrix \(A\) is called semipositive (SP) if there exists a vector \(x>0\) such that \(Ax>0\). For such class of matrices, the authors study the optimization problems \[ e(A) = \inf_{x>0, Ax>0\atop \prod_{i=1}^n x_i=1} \prod_{i=1}^n (Ax)_i \] and \[ E(A) = \sup_{x>0, Ax>0\atop \prod_{i=1}^n x_i=1} \prod_{i=1}^n (Ax)_i. \] The terms \(e(A)\) and \(E(A)\) are referred to as equilibrants of the matrix \(A\).