Theorems of Perron-Frobenius type for matrices without sign restrictions

AbstractThe paper attempts to solve a problem which was called a “challenge for the future” in Linear Algebra Appl. We define and investigate a new quantity for real matrices, the sign-real spectral radius, and derive various characterizations, bounds, and properties of it. In certain aspects our quantity shows similar behavior to the Perron root of a nonnegative matrix. It is shown that our quantity also has intimate connections to the componentwise distance to the nearest singular matrix. Relations to the Perron root of the (entrywise) absolute value of the matrix and to the μ-number are given as well

Improved Solution Enclosures for Over- and Underdetermined Interval Linear Systems

Abstract. In this paper we discuss an inclusion method for solving rectangular (over- and under-determined) dense linear systems where the input data are uncertain and vary within given intervals. An improvement of the quality of the solution enclosures is described for both independent and parameter dependent input intervals. A fixed-point algorithm with result verification that exploits the structure of the problems to be solved is given. Mathematica functions for solving the discussed rectangular problems are developed and presented. Numerical examples illustrate the advantages of the proposed improved approach.