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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: The first counterexample to this Finiteness conjecture was given in 2002 by T. Bousch and J. Mairesse and their proof was based on measure-theoretical ideas as discussed by the authors.
Abstract: In 1995 J.C. Lagarias and Y. Wang conjectured that the generalized spectral radius of a finite set of square matrices can be attained on a finite product of matrices. The first counterexample to this Finiteness Conjecture was given in 2002 by T. Bousch and J. Mairesse and their proof was based on measure-theoretical ideas. In 2003 V.D. Blondel, J. Theys and A.A. Vladimirov proposed another proof of a counterexample to the Finiteness Conjecture which extensively exploited combinatorial properties of permutations of products of positive matrices.
In the control theory, so as in the general theory of dynamical systems, the notion of generalized spectral radius is used basically to describe the rate of growth or decrease of the trajectories generated by matrix products. In this context, the above mentioned methods are not enough satisfactory (from the point of view of the author, of course) since they give no description of the structure of the trajectories with the maximal growing rate (or minimal decreasing rate).
In connection with this, in 2005 the author presented one more proof of the counterexample to the Finiteness Conjecture fulfilled in the spirit of the theory of dynamical systems. Unfortunately, the developed approach did not cover the class of matrices considered by Blondel, Theys and Vladimirov. The goal of the present paper is to compensate for this deficiency in the previous approach.
63 citations
TL;DR: It is shown that the tests proposed by Srivastava (2005) for the above three problems are robust under the non-normality assumption made in this article irrespective of whether [email protected]?p or N>=p, but (N,p)->~, and N/p may go to zero or infinity.
62 citations
01 Apr 1970
TL;DR: In this paper, it was shown that, given a complex square matrix A all of whose leading principal minors are nonzero, there is a diagonal matrix D such that the product DA of the two matrices has all its characteristic roots positive and simple.
Abstract: In this paper it is shown that, given a complex square matrix A all of whose leading principal minors are nonzero, there is a diagonal matrix D such that the product DA of the two matrices has all its characteristic roots positive and simple. This result is already known for real A, but two new proofs for this case are given here.
62 citations
TL;DR: A necessary and sufficient condition for all principal minors of a square matrix to be positive is given in this article, and a special subclass of such matrices called quasidominant matrices is also examined.
62 citations
TL;DR: In this article, the authors summarize and extend some of the applicable solution procedures through a systematic use of operators which convert the matrix equation to vector and dimension-reduced vector forms.
62 citations