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Square matrix
About: Square matrix is a research topic. Over the lifetime, 5000 publications have been published within this topic receiving 92428 citations.
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TL;DR: In this article, the characteristic roots of a square matrix are defined in terms of Hermitian matrices and a generalization of the result is obtained that covers cases of matrices B whether B does or does not converge to 0, except for very special matrices.
Abstract: If B is a square matrix, then it is known that a necessary and sufficient condition that J^n B=0, is that the characteristic roots of Bare all of modulus less than unity. An alternative condition is given in this paper, in terms of Hermitian matrices. Further, a generalization of the result is obtained that covers cases of matrices B whether B does or does not converge to 0, except for very special matrices.
85 citations
TL;DR: Using the shifted number system the high-order lifting and integrality certification techniques of Storjohann 2003 for polynomial matrices are extended to the integer case.
85 citations
TL;DR: In this paper, the authors proposed to reduce the N-wave interaction equations into finite-dimensional Liouville integrable systems, where the potentials resulting from the symmetry constraints give rise to involutive solutions.
Abstract: Binary symmetry constraints of the N-wave interaction equations in 1+1 and 2+1 dimensions are proposed to reduce the N-wave interaction equations into finite-dimensional Liouville integrable systems. A new involutive and functionally independent system of polynomial functions is generated from an arbitrary order square matrix Lax operator and used to show the Liouville integrability of the constrained flows of the N-wave interaction equations. The constraints on the potentials resulting from the symmetry constraints give rise to involutive solutions to the N-wave interaction equations, and thus the integrability by quadratures are shown for the N-wave interaction equations by the constrained flows.
85 citations
TL;DR: It is shown that this method yields very satisfactory analyses for journal and national citation data, enabling the members of the set to be assigned measures of size, quality and self-interest and a fuzzy set of clustered members from which all data may be derived.
Abstract: A method is explained for analysing square matrices of statistics giving transactions between each member of a set of nations, papers, journals, etc. In general self-transactions are different in kind to other exchanges of money, citations, etc., and a special method is given to compute row and column coefficients without relying on the diagonal elements. It is shown that this method yields very satisfactory analyses for journal and national citation data, enabling the members of the set to be assigned measures of size, quality and self-interest and a fuzzy set of clustered members from which all data may be derived.
85 citations
TL;DR: In a recent paper as mentioned in this paper, the authors have given an admirable summary of the theory of permanents of square matrices, especially with regard to inequalities satisfied by the permanent and the determinant.
Abstract: In a recent paper [1] Marcus and Minc have given an admirable summary of the theory of permanents of square matrices, especially with regard to inequalities satisfied by the permanent. Our knowledge of permanents is certainly meager compared to that for determinants, largely for the reason that while the former differs from the latter only in the replacement of minus signs by plus signs, the determinant, and not the permanent, is invariant under unitary transformations.
85 citations