247,417 research outputs found
Some results about inverse-positive matrices
A nonsingular real matrix A is said to be inverse-positive if all the elements of its inverse are nonnegative. This class of matrices contains the M-matrices, from which inherit some of their properties and applications, especially in economy and in the description of iterative methods for solving nonlinear systems. In this paper we present some new characterizations for inverse-positive matrices and we analyze when this concept is preserved by the sub-direct sum of matrices. © 2011 Elsevier Inc. All rights reserved.This research was supported by Ministerio de Ciencia y Tecnologia MTM2010-18539.Abad RodrÃguez, MF.; Gasso Matoses, MT.; Torregrosa Sánchez, JR. (2011). Some results about inverse-positive matrices. Applied Mathematics and Computation. 218(1):130-139. https://doi.org/10.1016/j.amc.2011.05.063S130139218
Completion of partial operator matrices
This work concerns completion problems for partial operator matrices. A partial matrix is an m-by-n array in which some entries are specified and the remaining are unspecified. We allow the entries to be operators acting between corresponding vector spaces (in general, bounded linear operators between Hilbert spaces). Graphs are associated with partial matrices. Chordal graphs and directed graphs with a perfect edge elimination scheme play a key role in our considerations. A specific choice for the unspecified entries is referred to as a completion of the partial matrix. The completion problems studied here involve properties such as: zero-blocks in certain positions of the inverse, positive (semi)definitness, contractivity, or minimum negative inertia for Hermitian operator matrices. Some completion results are generalized to the case of combinatorially nonsymmetric partial matrices. Several applications including a maximum entropy result and determinant formulae for matrices with sparse inverses are given.;In Chapter II we treat completion problems involving zero-blocks in the inverse. Our main result deals with partial operator matrices R, for which the directed graph is associated with an oriented tree. We prove that under invertibility conditions on certain principal minors, R admits a unique invertible completion F such that {dollar}(F\sp{lcub}-1{rcub})\sb{lcub}ij{rcub}{dollar} = 0 whenever {dollar}R\sb{lcub}ij{rcub}{dollar} is unspecified.;Chapter III treats positive semidefinite and Hermitian completions. In the case of partial positive operator matrices with a chordal graph, a maximum entropy principle is presented, generalizing the maximum determinant result in the scalar case. We obtain a linear fractional transform parametrization for the set of all positive semidefinite completions for a generalized banded partial matrix. We also give an inertia formula for Hermitian operator matrices with sparse inverses.;In Chapter IV prior results are applied to obtain facts about contractive and linearly constrained completion problems. The solution to a general n-by-n strong-Parrott type completion problem is the main result. We prove necessary and sufficient conditions for the existence of a solution as well as a cascade transform parametrization for the set of all solutions.;Chapter V extends the results in Chapter II and III to prove determinant formulae for matrices with sparse inverses. Several ideas from graph theory are used. An inheritance principle for chordal graphs is also presented
The NIEP
The nonnegative inverse eigenvalue problem (NIEP) asks which lists of
complex numbers (counting multiplicity) occur as the eigenvalues of some
-by- entry-wise nonnegative matrix. The NIEP has a long history and is a
known hard (perhaps the hardest in matrix analysis?) and sought after problem.
Thus, there are many subproblems and relevant results in a variety of
directions. We survey most work on the problem and its several variants, with
an emphasis on recent results, and include 130 references. The survey is
divided into: a) the single eigenvalue problems; b) necessary conditions; c)
low dimensional results; d) sufficient conditions; e) appending 0's to achieve
realizability; f) the graph NIEP's; g) Perron similarities; and h) the
relevance of Jordan structure
Optimal Linear Shrinkage Estimator for Large Dimensional Precision Matrix
In this work we construct an optimal shrinkage estimator for the precision
matrix in high dimensions. We consider the general asymptotics when the number
of variables and the sample size so
that . The precision matrix is estimated
directly, without inverting the corresponding estimator for the covariance
matrix. The recent results from the random matrix theory allow us to find the
asymptotic deterministic equivalents of the optimal shrinkage intensities and
estimate them consistently. The resulting distribution-free estimator has
almost surely the minimum Frobenius loss. Additionally, we prove that the
Frobenius norms of the inverse and of the pseudo-inverse sample covariance
matrices tend almost surely to deterministic quantities and estimate them
consistently. At the end, a simulation is provided where the suggested
estimator is compared with the estimators for the precision matrix proposed in
the literature. The optimal shrinkage estimator shows significant improvement
and robustness even for non-normally distributed data.Comment: 26 pages, 5 figures. This version includes the case c>1 with the
generalized inverse of the sample covariance matrix. The abstract was updated
accordingl
Fibonacci numbers and orthogonal polynomials
We prove that the sequence of reciprocals of the
Fibonacci numbers is a moment sequence of a certain discrete probability, and
we identify the orthogonal polynomials as little -Jacobi polynomials with
. We prove that the corresponding kernel
polynomials have integer coefficients, and from this we deduce that the inverse
of the corresponding Hankel matrices have integer entries. We
prove analogous results for the Hilbert matrices.Comment: A note dated June 2007 has been added with some historical comments.
Some references have been added and complete
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