Playing with positive matrices: efficiency of an initial state (Q2912630)

This paper deals with the asymptotic behavior of certain dynamical systems in \(\mathbb{R}^k_+=[0, +\infty)^k\) defined by \(v_{n+1}=Av_n\), where \(A\) is a \(k \times k\) nonnegative matrix, called transition matrix, and \(v_n\) is the state vector of the system at the observation period \(n\).NEWLINENEWLINEIf \(\lambda_p\) denotes the spectral radius of the \(k \times k\) nonnegative matrix \(A\), it is known that \(\lambda_p\) is an eigenvalue of \(A\) and \(\operatorname{ker}(A-\lambda_pI_k)\cap \mathbb{R}^k_+ \neq \emptyset\). It is said that \(\lambda_p\) is dominant if it is a simple root of the characteristic polynomial. It is also known that given an initial state vector \(v_0 \in \mathbb{R}^k_+\) there exists a constant \(c(v_0) \in \mathbb{R}_+\) such that, if \(v_n=A^nv_0\), \(n \in \mathbb{N}\), is a solution of the dynamical system, then NEWLINE\[NEWLINE v_n \rightarrow c(v_0) \lambda_p^n p, \;\;n \rightarrow +\infty, NEWLINE\]NEWLINE where \(p\) is the unique eigenvector whose components sum 1. \(c(v_0)\) is called efficiency of state \(v_0\). The vector efficiency of the model is \(c=(c_1,c_2, \ldots,c_k)^T\), where \(c_i=c(e_i), i=1,2,\ldots,k\), and \(\{ e_1, e_2, \ldots, e_k \}\) the canonical base of \(\mathbb{R}^k\)NEWLINENEWLINEIn this work, the authors study the vector efficiency of the model. When \(\lambda_p\) is dominant they prove that this vector is the eigenvector of \(A^T\) associated to \(\lambda_p\) such that \(\langle c,p \rangle=1\).NEWLINENEWLINEIf \(\lambda_p\) is not dominant, but \(\dim{(\operatorname{ker}(A-\lambda_p I))}=1\), the authors establish that \(c\) is an eigenvector of \(A^T\) associated to \(\lambda_p\).NEWLINENEWLINEFor the entire collection see [Zbl 1243.00023].