Let P be a stochastic matrix. We give a necessary and sufficient condition for the existence of the limt -> ∞ ppt from which follows the classical regularity conditions. Another regularity condition based in the Banach point fix theorem is also given.
A graph is nonsingular if its adjacency matrix is nonsingular. The inverse of a nonsingular graph is a graph whose adjacency matrix is similar to via a particular type of similarity. Let denote the class of connected bipartite graphs with unique perfect matchings. Tifenbach and Kirkland (2009) characterized the unicyclic graphs in which possess unicyclic inverses. We present a characterization of unicyclic graphs in which possess bicyclic inverses.
Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the sth secant variety of the d-uple Veronese embedding of into but that its minimal decomposition as a sum of dth powers of linear forms requires more than s summands. We show that if s ≤ d then F can be uniquely written as , where are linear forms with t ≤ (d-1)/2, and Q is a binary form such that with ’s linear forms and ’s forms...
We establish a unique factorization result into irreducibel elements in the partial semigroup of 2 × 2-matrices with entries in K[x] whose determinant is equal to 1, where K is a field, and where multiplication is defined as the usual matrix-multiplication if the degrees of the factors add up. This investigation is motivated by a result on matrices of entire functions.
Our motivation was a paper of 1991 indicating three special unitary matrices that map Hermitian Toeplitz matrices by similarity into real Toeplitz-plus-Hankel matrices. Generalizing this result, we give a complete description of unitary similarity automorphisms of the space of Toeplitz-plus-Hankel matrices.
We show that any compact semigroup of n × n matrices is similar to a semigroup bounded by √n. We give examples to show that this bound is best possible and consider the effect of the minimal rank of matrices in the semigroup on this bound.
We show that any compact semigroup of positive n × n matrices is similar (via a positive diagonal similarity) to a semigroup bounded by √n. We give examples to show this bound is best possible. We also consider the effect of additional conditions on the semigroup and obtain improved bounds in some cases.
We study the universality of the local eigenvalue statistics of Gaussian divisible Hermitian Wigner matrices. These random matrices are obtained by adding an independent GUE matrix to an Hermitian random matrix with independent elements, a Wigner matrix. We prove that Tracy–Widom universality holds at the edge in this class of random matrices under the optimal moment condition that there is a uniform bound on the fourth moment of the matrix elements. Furthermore, we show that universality holds...