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On the inertia sets of some symmetric sign patterns

C. M. da Fonseca (2006)

Czechoslovak Mathematical Journal

A matrix whose entries consist of elements from the set { + , - , 0 } 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.

On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution

A. Pajor, L. Pastur (2009)

Studia Mathematica

We consider n × n real symmetric and hermitian random matrices Hₙ that are sums of a non-random matrix H ( 0 ) 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 H ( 0 ) and the distribution of amplitudes converge weakly, then the distribution of eigenvalues of Hₙ converges...

On the matrices of central linear mappings

Hans Havlicek (1996)

Mathematica Bohemica

We show that a central linear mapping of a projectively embedded Euclidean n -space onto a projectively embedded Euclidean m -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 2 m - n + 1 . This matrix is arising, by a simple manipulation, from a matrix describing the given mapping in terms of homogeneous Cartesian coordinates.

On the optimality and sharpness of Laguerre's lower bound on the smallest eigenvalue of a symmetric positive definite matrix

Yusaku Yamamoto (2017)

Applications of Mathematics

Lower bounds on the smallest eigenvalue of a symmetric positive definite matrix A m × m play an important role in condition number estimation and in iterative methods for singular value computation. In particular, the bounds based on Tr ( A - 1 ) and Tr ( A - 2 ) have attracted attention recently, because they can be computed in O ( m ) operations when A 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...

On the separation of eigenvalues by the permutation group

Grega Cigler, Marjan Jerman (2014)

Special Matrices

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.

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