Topic
Spectrum of a matrix
About: Spectrum of a matrix is a research topic. Over the lifetime, 1064 publications have been published within this topic receiving 19841 citations. The topic is also known as: matrix spectrum.
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TL;DR: In this paper, the stabilization of the linear periodic discrete-time system through the use of linear periodic state-variable feedback has been studied and it is shown that complete reachability of an open-loop system is equivalent to the possibility of assigning an arbitrary set of the eigenvalues to Y(τ, 0) by choosing a suitable state feedback.
Abstract: This paper considers the stabilization of the linear periodic discrete-time system through the use of linear periodic state-variable feedback. Let the transition matrix of the closed-loop system be Y(t, s). Then the stability of the closed-loop system depends on eigenvalues of Y(τ,0), where τ is period. It is shown that complete reachability of an open-loop system is equivalent to the possibility of assigning an arbitrary set of the eigenvalues to Y(τ, 0) by choosing a suitable state feedback.
98 citations
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TL;DR: In this paper, the eigenvalue problem Ax = λBx is shown to have a complete system of eigenvectors and that its eigenvalues are real.
98 citations
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TL;DR: The main hypothesis is a coercivity condition on the contrast that must hold only in a neighborhood of the boundary that proves that the interior transmission operator has upper triangular compact resolvent and the spectrum is discrete and the generalized eigenspaces are finite-dimensional.
Abstract: Transmission eigenvalues are points in the spectrum of the interior transmission operator, a coupled $2 \times 2$ system of elliptic partial differential equations, where one unknown function must satisfy two boundary conditions and the other must satisfy none. We show that the interior transmission eigenvalues are discrete and depend continuously on the contrast by proving that the interior transmission operator has upper triangular compact resolvent, and that the spectrum of these operators share many of the properties of operators with compact resolvent. In particular, the spectrum is discrete and the generalized eigenspaces are finite-dimensional. Our main hypothesis is a coercivity condition on the contrast that must hold only in a neighborhood of the boundary.
98 citations
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TL;DR: In this paper, the concept of left and right eigenvalues for a quaternionic matrix was introduced, and the properties, quantities and relationship of these eigenvectors were investigated.
97 citations
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TL;DR: In this paper, an unfolding procedure for complex eigenvalues was introduced and applied to data from lattice QCD at finite chemical potential to construct the nearest-neighbor spacing distribution of adjacent eigen values in the complex plane.
Abstract: In quantum chromodynamics (QCD) at nonzero chemical potential, the eigenvalues of the Dirac operator are scattered in the complex plane. Can the fluctuation properties of the Dirac spectrum be described by universal predictions of non-Hermitian random matrix theory? We introduce an unfolding procedure for complex eigenvalues and apply it to data from lattice QCD at finite chemical potential $\mu$ to construct the nearest-neighbor spacing distribution of adjacent eigenvalues in the complex plane. For intermediate values of $\mu$, we find agreement with predictions of the Ginibre ensemble of random matrix theory, both in the confinement and in the deconfinement phase.
97 citations