Some unique properties of eigenvector centrality
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We show that for any two [Formula: see text] matrices [Formula: see text] and [Formula: see text] we have the inequality [Formula: see text] where [Formula: see text] and [Formula: see text] denote the decreasingly ordered singular values and eigenvalues of [Formula: see text]. As an application, we show that for [Formula: see text] real positive definite matrices the symplectic eigenvalues [Formula: see text] under some special conditions, satisfy the inequality [Formula: see text].
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The purpose of this paper is to locate and estimate the left eigenvalues of quaternionic matrices. We present some distributiontheorems for the left eigenvalues of square quaternionic matrices based on the generalized Gerschgorin theoremand generalized Brauer theorem.
Quaternionic representation
Matrix (chemical analysis)
Normal matrix
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Detection of different kinds of anomalous behaviors originating from negative ties among actors in online social networks is an unexplored area requiring extensive research. Due to increase in social crimes such as masquerading, bullying, etc., identification and analysis of these activities has become need of the hour. Approaches from two separate, yet, similar research areas, i.e. anomaly detection and negative tie analysis, can be clubbed together to identify negative anomalous nodes. Use of best measures from centrality based (negative ties) and structure based approaches (anomaly detection) can help us identify and analyze the negative ties more efficiently. A comparative analysis has been performed to detect the negative behaviors in online networks using different centrality measures and their relationship in curve fitting anomaly detection techniques. From results it is observed that curve fitting analysis of centrality measures relationship performs better than independent analysis of centrality measures for detecting negative anomalous nodes.
Identification
Anomaly (physics)
Network Analysis
Social Network Analysis
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We consider the problem of determining the best possible bounds on the eigenvalues of an n th order positive definite matrix B , when the determinant ( D ) and trace ( T ) are given. A large variety of bounds on the eigenvalues are known when different information concerning B is available (see, for example, [1], [2]). Since D and T simply provide the geometric mean and arithmetic mean of the positive, real eigenvalues of B , the solution to the problem involves certain inequalities satisfied by these means (see [3] for such inequalities in a more general setting). A related problem in which the largest and smallest eigenvalue are known, and inequalities involving D and T are obtained, is described in [4].
TRACE (psycholinguistics)
Matrix (chemical analysis)
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Algebraic connectivity
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It is shown for coefficient matrices of Russell-Rao coefficients and two asymmetric Dice coefficients that ordinal information on a latent variable model can be obtained from the eigenvector corresponding to the largest eigenvalue.
Similarity (geometry)
Dice
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It is known [10, 11] that if T is an integral operator with an extended totally positive kernel, then T has a countably infinite family of simple, positive eigenvalues. We prove a similar result for a rather larger class of kernels and, writing the eigenvalues of T in decreasing order as (λn)n∈N, we use results obtained in [4] and [5] to give a formula for the ratio λn+1/λn analogous to that given in [3] for the case of a strictly totally positive matrix, and to the spectral radius formula r ( T ) = lim n → ∞ ‖ T n ‖ 1 / n = inf n ∈ N ‖ T n ‖ 1 / n . This may be regarded as a generalisation of inequalities due to Hopf [8, 9]. 1991 1991 Mathematics Subject Classification 47G10, 47B65.
Spectral Radius
Operator (biology)
Mathematics Subject Classification
Matrix (chemical analysis)
Kernel (algebra)
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Eigenfunction
Sturm–Liouville theory
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