Convergence of products of matrices in projective spaces
This paper deals with convergence of products of positive matrices in projective spaces. For \(\mathbb R_+=(0,\infty)\) the author denotes by \(\mathbb R_+^n\), \(M_n(\mathbb R_+)\) the cone of positive vectors and the semialgebra of positive matrices. Given \(A_k \in M_n(\mathbb R_+)\), \(k \in N\), a sequence of positive matrices which converges a positive matrix \(A \in M_n(\mathbb R_+)\), the main results of this paper show that the product \[ \frac{A_k A_{k-1}\cdots A_2 A_1}{\| A_k A_{k-1}\cdots A_2 A_1 \| } \] converges to a positive matrix \(E\) such that \(\| E\| =1\). Furthermore, \[ E=uw^T, \;u,w \in \mathbb R_+^n, \;Au=\rho(A)u. \] In addition, if each \(A_k\) is nonsingular and \(A\) has exactly one simple eigenvalue \(\lambda\) of the maximal modulus with the corresponding eigenvector \(u\), then \(e^{\sqrt{-1}\theta_k}\frac{A_k A_{k-1}\cdots A_2 A_1}{\| A_k A_{k-1}\cdots A_2 A_1 \| }\), \(\theta_k \in \mathbb{R}\), converges to a rank one matrix \(uw^T\). Finally, the author extends the previous result to strictly totally positive matrices of order \(p\).
scientific article; zbMATH DE number 5014280