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Matrix Shanks Transformations
No full textShanks\u27 transformation is a well know sequence transformation for accelerating the convergence of scalar sequences. It has been extended to the case of sequences of vectors and sequences of square matrices satisfying a linear difference equation with scalar coefficients. In this paper, a more general extension to the matrix case where the matrices can be rectangular and satisfy a difference equation with matrix coefficients is proposed and studied. In the particular case of square matrices, the new transformation can be recursively implemented by the matrix -algorithm of Wynn. Then, the transformation is related to matrix Pad\\u27{e}-type and Pad\\u27{e} approximants. Numerical experiments showing the interest of this transformation end the paper
Alpha Adjacency: A generalization of adjacency matrices
No full textB. Shader and W. So introduced the idea of the skew adjacency matrix. Their idea was to give an orientation to a simple undirected graph G from which a skew adjacency matrix S(G) is created. The -adjacency matrix extends this idea to an arbitrary field F. To study the underlying undirected graph, the average -characteristic polynomial can be created by averaging the characteristic polynomials over all the possible orientations. In particular, a Harary-Sachs theorem for the average-characteristic polynomial is derived and used to determine a few features of the graph from the average-characteristic polynomial
Rape is Not a Contract: Recognizing the Fundamental Difficulties in Applying Economic Theories of Jurisprudence to Criminal Sexual Assault
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Wyoming is More Likely Than Not Behind in Guardianship Proceedings: The Unconstitutional Standard for Guardianship Under Wyoming Statute § 3-2-104
No full textOn the Perron-Frobenius Theory of matrices and Eventually Exponentially Nonnegative Matrices
No full textmatrix is a matrix of the form , where and is an eventually nonnegative matrix. In this paper, matrices concerning the Perron-Frobenius theory are studied. Specifically, sufficient and necessary conditions for an matrix to have positive left and right eigenvectors corresponding to its eigenvalue with smallest real part without considering or not if are stated and proven. Moreover, analogous conditions for eventually nonnegative matrices or matrices to have all the non Perron eigenvectors or generalized eigenvectors not being nonnegative are studied. Then, equivalent properties of eventually exponentially nonnegative matrices and matrices are presented. Various numerical examples are given to support our theoretical findings
In-sphere property and reverse inequalities for matrix means
No full textThe in-sphere property for matrix means is studied. It is proved that the matrix power mean satisfies in-sphere property with respect to the Hilbert-Schmidt norm. A new characterization of the matrix arithmetic mean is provided. Some reverse AGM inequalities involving unitarily invariant norms and operator monotone functions are also obtained
Vector Spaces of Generalized Linearizations for Rectangular Matrix Polynomials
No full textThe complete eigenvalue problem associated with a rectangular matrix polynomial is typically solved via the technique of linearization. This work introduces the concept of generalized linearizations of rectangular matrix polynomials. For a given rectangular matrix polynomial, it also proposes vector spaces of rectangular matrix pencils with the property that almost every pencil is a generalized linearization of the matrix polynomial which can then be used to solve the complete eigenvalue problem associated with the polynomial. The properties of these vector spaces are similar to those introduced in the literature for square matrix polynomials and in fact coincide with them when the matrix polynomial is square. Further, almost every pencil in these spaces can be `trimmed\u27 to form many smaller pencils that are strong linearizations of the matrix polynomial which readily yield solutions of the complete eigenvalue problem for the polynomial. These linearizations are easier to construct and are often smaller than the Fiedler linearizations introduced in the literature for rectangular matrix polynomials. Additionally, a global backward error analysis applied to these linearizations shows that they provide a wide choice of linearizations with respect to which the complete polynomial eigenvalue problem can be solved in a globally backward stable manner