3,964 research outputs found
Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X+AtX-1=I
Matrices;numerieke wiskunde
On the existence of a positive definite solution of the matrix equation X+AXA=I
Matrices;mathematics
Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X+A*X-1A=Q
Matrices;numerieke wiskunde
Stability and Stabilization of the Wave Model.
The stability properties of 2-D systems are an important aspect of the design of acoustic, seismic, image and sonar signal processors. This research utilizes the Wave model format to transport 1-D stability techniques to the 2-D setting. The research studies stability through multistep growth bounds on the Wave state. The use of Lyapunov theory is also considered. The research considers also the problem of stabilizing a 2-D system using state and/or output information feedback to interior and/or boundary controls. Finally the problem of observer design for 2-D systems is considered, with the new stability criteria being used to assure observer/system convergence. New results based on symmetrizability are also discussed. The principal results are illustrated by a number of examples. The results are also interpreted in the context of other contemporary local state models
Spectral Theory of Sparse Non-Hermitian Random Matrices
Sparse non-Hermitian random matrices arise in the study of disordered
physical systems with asymmetric local interactions, and have applications
ranging from neural networks to ecosystem dynamics. The spectral
characteristics of these matrices provide crucial information on system
stability and susceptibility, however, their study is greatly complicated by
the twin challenges of a lack of symmetry and a sparse interaction structure.
In this review we provide a concise and systematic introduction to the main
tools and results in this field. We show how the spectra of sparse
non-Hermitian matrices can be computed via an analogy with infinite dimensional
operators obeying certain recursion relations. With reference to three
illustrative examples --- adjacency matrices of regular oriented graphs,
adjacency matrices of oriented Erd\H{o}s-R\'{e}nyi graphs, and adjacency
matrices of weighted oriented Erd\H{o}s-R\'{e}nyi graphs --- we demonstrate the
use of these methods to obtain both analytic and numerical results for the
spectrum, the spectral distribution, the location of outlier eigenvalues, and
the statistical properties of eigenvectors.Comment: 60 pages, 10 figure
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