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Descripción
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| Our interest in nearly reducible matrices arises from our prior interest in %We are previously interested in the following nonnegative inverse eigenvalue problem \cite{To}: given $k_1, k_2,\dots, k_n$ real numbers, find necessary and sufficient conditions for the existence of a nonnegative matrix $A$ of order $n$ with characteristic polynomial $x^n+k_1x^{n-1}+ k_2 x^{n-2}+\dots+ k_n$. The matricial realizations of the characteristic polynomial can be described in terms of irreducible matrices by means of their normal forms of Frobenius. The class of irreducible matrices can easily be reduced to the class of nearly reducible (minimal irreducible) matrices, so we are interested in any theoretical or constructive characterization of these classes of matrices. We introduce an adequate concept of expansion of a $(0,1)$-square matrix to obtain a sequential construction of nearly reducible matrices. We characterize the class of nearly reducible matrices whose expansion preserves the property of minimality. We prove that every nearly reducible matrix of order $n\geq 2$ is the expansion of a nearly reducible matrix of order $n-1$ and we give sequentially generative procedures for the constructive characterization of the classes of nearly reducible matrices. We describe algorithms to compute nearly reducible matrices (up to permutational congruency) and their isospectral classes. With respect to our initial motivation of the nonnegative inverse eigenvalue problem, the problem {\it ``which monic polynomials of degree $n$ with integral coefficients are the characteristic polynomials of nearly reducible matrices of order $n$''} has been solved in the sense that the above algorithms allow the class of characteristic polynomials of the nearly reducible matrices of order $n$ and the sets of nearly reducible matrices with equal characteristic polynomial to be catalogued \cite{GarMa}. | |
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Internacional
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Si |
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Nombre congreso
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Congreso de la Real Sociedad Matemática Española 2011 |
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Tipo de participación
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960 |
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Lugar del congreso
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Avila (España) |
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Revisores
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Si |
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ISBN o ISSN
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0000-0000 |
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DOI
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Fecha inicio congreso
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01/05/2011 |
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Fecha fin congreso
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05/05/2011 |
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Desde la página
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46 |
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Hasta la página
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47 |
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Título de las actas
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Congreso de la Real Sociedad Matemática Española 2011 - posters |