An invertible matrix is called a Perron similarity if one of its columns and the corresponding row inverse are both nonnegative or nonpositive. Such matrices relevance import in study eigenvalue problem. In this work, Kronecker products similarities examined used to construct ideal all whose rows extremal.

In this thesis we present the formalization of three principal results that are the Jordan normal form of a matrices, the Bolzano-Weierstraß theorem, and the Perron-Frobenius theorem. To formalize the Jordan normal form, we introduce many concepts of linear algebra like block diagonal matrices, companion matrices, invariant factors, ... The formalization of Bolzano-Weierstraß theorem needs to d...

M. I. Petaev, A. Meibom, C. Perron, and B. Zanda, Dept. of the Geophysical Sciences, University of Chicago (5734 S. Ellis Ave., Chicago, IL 60637 USA; [email protected]), Dept. of Earth and Planetary Sciences, Harvard University (20 Oxford St., Cambridge, MA 02138 USA; [email protected]), Dept. of Geological and Environmental Sciences, Stanford University (Building 320, Lomita Mall,...

A parallel algorithm is presented for computing the group inverse of a singular M–matrix of the form A = I − T , where T ∈ Rn×n is irreducible and stochastic. The algorithm is constructed in the spirit of Meyer’s Perron complementation approach to computing the Perron vector of an irreducible nonnegative matrix. The asymptotic number of multiplication operations that is necessary to implement t...

We establish a stochastic nonlinear analogue of the PerronFrobenius theorem on eigenvalues and eigenvectors of positive matrices. The result is formulated in terms of an automorphism T of a probability space (Ω,F , P ) and a random mapping D(ω, ·) : R+ → R+. Under assumptions of monotonicity and homogeneity of D(ω, ·), we prove the existence of scalar and vector measurable functions α(ω) > 0 an...

The main results of this paper are generalizations of classical results from the numerical range to the block numerical range. A different and simpler proof for the Perron-Frobenius theory on the block numerical range of an irreducible nonnegative matrix is given. In addition, the Wielandt's lemma and the Ky Fan's theorem on the block numerical range are extended.

We analyse dynamical properties of the negative beta transformation, which has been studied recently by Ito and Sadahiro. Contrary to the classical beta transformation, the density of the absolutely continuous invariant measure of the negative beta transformation may be zero on certain intervals. By investigating this property in detail, we prove that the (−β)-transformation is exact for all β ...