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: This paper derives a new test matrix for S-parameter models which is only half the size of the Hamiltonian matrix, leading to savings in the eigenvalue computation time by a factor of nearly eight.
Abstract: Rational models must be passive in order to ensure stable time domain simulations. The assessment of passivity properties is usually done via a Hamiltonian matrix that is associated with the state-space model, allowing precise characterization of passivity violations from its imaginary eigenvalues. The calculation of eigenvalues can be time consuming for large models as the matrix size is equal to twice the number of model states. In this paper, we derive for S-parameter models a new test matrix which is only half the size of the Hamiltonian matrix. This leads to savings in the eigenvalue computation time by a factor of nearly eight. The new test matrix takes into account that the model is symmetrical, in pole-residue form. Its application is demonstrated by three examples: a microwave filter, a package, and a synthetic model.
76 citations
TL;DR: In this article, a new upper bound on the sum of the k largest Laplacian eigenvalues of every n-vertex tree was established, where k ∈{1, 1, n, n}.
75 citations
TL;DR: Benaych-Georges, Bordenave and Knowles as mentioned in this paper showed that the extreme eigenvalues of Erdős-Renyi graphs exhibit a novel behavior which in particular rules out their convergence to a non-degenerate point process.
Abstract: We consider inhomogeneous Erdős–Renyi graphs. We suppose that the maximal mean degree $d$ satisfies $d\ll\log n$. We characterise the asymptotic behaviour of the $n^{1-o(1)}$ largest eigenvalues of the adjacency matrix and its centred version. We prove that these extreme eigenvalues are governed at first order by the largest degrees and, for the adjacency matrix, by the nonzero eigenvalues of the expectation matrix. Our results show that the extreme eigenvalues exhibit a novel behaviour which in particular rules out their convergence to a nondegenerate point process. Together with the companion paper [Benaych-Georges, Bordenave and Knowles (2017)], where we analyse the extreme eigenvalues in the complementary regime $d\gg\log n$, this establishes a crossover in the behaviour of the extreme eigenvalues around $d\sim\log n$. Our proof relies on a tail estimate for the Poisson approximation of an inhomogeneous sum of independent Bernoulli random variables, as well as on an estimate on the operator norm of a pruned graph due to Le, Levina, and Vershynin from [Random Structures Algorithms 51 (2017) 538–561].
74 citations
TL;DR: A method is described for the computation of rigorous error bounds for multiple or nearly multiple eigenvalues, and for a basis of the corresponding invariant subspaces, based on a quadratically convergent Newton-like method.
74 citations
TL;DR: In this article, the authors obtained complex nonlinear integral equations for the two asymptotically degenerate maximum eigenvalues of the transfer matrix of the eight-vertex model.
Abstract: We obtain complex nonlinear integral equations for the two asymptotically degenerate maximum eigenvalues of the transfer matrix of the eight-vertex model. These are exact for a lattice of a finite numberN of columns. Solving the equations recursively gives an expansion of the eigenvalues aboutN = ∞. Thus we can obtain the interfacial tension of the model, as well as rederiving our previous result for the free energy.
74 citations