A matrix whose entries consist of elements from the set is a sign pattern matrix. Using a linear algebra theoretical approach we generalize of some recent results due to Hall, Li and others involving the inertia of symmetric tridiagonal sign matrices.
We consider n × n real symmetric and hermitian random matrices Hₙ that are sums of a non-random matrix and of mₙ rank-one matrices determined by i.i.d. isotropic random vectors with log-concave probability law and real amplitudes. This is an analog of the setting of Marchenko and Pastur [Mat. Sb. 72 (1967)]. We prove that if mₙ/n → c ∈ [0,∞) as n → ∞, and the distribution of eigenvalues of and the distribution of amplitudes converge weakly, then the distribution of eigenvalues of Hₙ converges...
We show that a central linear mapping of a projectively embedded Euclidean -space onto a projectively embedded Euclidean -space is decomposable into a central projection followed by a similarity if, and only if, the least singular value of a certain matrix has multiplicity . This matrix is arising, by a simple manipulation, from a matrix describing the given mapping in terms of homogeneous Cartesian coordinates.
Lower bounds on the smallest eigenvalue of a symmetric positive definite matrix play an important role in condition number estimation and in iterative methods for singular value computation. In particular, the bounds based on and have attracted attention recently, because they can be computed in operations when is tridiagonal. In this paper, we focus on these bounds and investigate their properties in detail. First, we consider the problem of finding the optimal bound that can be computed...
We consider the two-sided eigenproblem over max algebra. It is shown that any finite system of real intervals and points can be represented as spectrum of this eigenproblem.
Let A be an invertible 3 × 3 complex matrix. It is shown that there is a 3 × 3 permutation matrix P such that the product PA has at least two distinct eigenvalues. The nilpotent complex n × n matrices A for which the products PA with all symmetric matrices P have a single spectrum are determined. It is shown that for a n × n complex matrix [...] there exists a permutation matrix P such that the product PA has at least two distinct eigenvalues.