A result by Brauer shows how to modify one single eigenvalue of a matrix via a rank-one perturbat... more A result by Brauer shows how to modify one single eigenvalue of a matrix via a rank-one perturbation, without changing any of the remaining eigenvalues. This, together with the properties of real matrices with constant row sums, was exploited by the authors in a previous work in connection with the nonnegative inverse eigenvalue problem, obtaining conditions which are sufficient for the existence of an entrywise nonnegative matrix with prescribed spectrum. In this work we make use of Brauer’s Theorem again, to show that most of the previous results giving sufficient conditions for the real nonnegative inverse eigenvalue problem can be derived by using Brauer’s Theorem. Moreover, the technique is constructive, and there is an algorithmic procedure to construct a matrix realizing the spectrum. In particular, we show that if either Kellogg’s realizability criterion or Borobia’s realizability criterion is satisfied, then Soto’s realizability criterion is also satisfied. None of the conv...
A connection is established between the problem of characterizing all possible real spectra of en... more A connection is established between the problem of characterizing all possible real spectra of entrywise nonnegative matrices (the so-called real nonnegative inverse eigenvalue problem) and a combinatorial process consisting in repeated application of three basic manipulations on sets of real numbers. Given realizable sets (i.e., sets which are spectra of some nonnegative matrix), each of these three elementary transformations constructs a new realizable set. This defines a special kind of realizability, called C-realizability and this is closely related to the idea of compensation. After observing that the set of all C-realizable sets is a strict subset of the set of realizable ones, we show that it strictly includes, in particular, all sets satisfying several previously known sufficient realizability conditions in the literature. Furthermore, the proofs of these conditions become much simpler when approached from this new point of view. (A. Borobia), jmoro@math.uc3m.es (J. Moro), ...
Page 1. POSITIVE MATRICES WITH PRESCRIBED SINGULAR VALUES ∗ EMEDIN MONTA NO UNIVERSIDAD DE MAGAL... more Page 1. POSITIVE MATRICES WITH PRESCRIBED SINGULAR VALUES ∗ EMEDIN MONTA NO UNIVERSIDAD DE MAGALLANES, CHILE MARIO SALAS UNIVERSIDAD CAT ´OLICA DEL NORTE, CHILE and RICARDO L ...
Page 1. ELA EXTREME SPECTRA REALIZATION BY REAL SYMMETRIC TRIDIAGONAL AND REAL SYMMETRIC ARROW MA... more Page 1. ELA EXTREME SPECTRA REALIZATION BY REAL SYMMETRIC TRIDIAGONAL AND REAL SYMMETRIC ARROW MATRICES∗ HUBERT PICKMANN, JUAN C. EGANA, AND RICARDO L. SOTO§ Abstract. We consider ...
A result by Brauer shows how to modify one single eigenvalue of a matrix via a rank-one perturbat... more A result by Brauer shows how to modify one single eigenvalue of a matrix via a rank-one perturbation, without changing any of the remaining eigenvalues. This, together with the properties of real matrices with constant row sums, was exploited by the authors in a previous work in connection with the nonnegative inverse eigenvalue problem, obtaining conditions which are sufficient for the existence of an entrywise nonnegative matrix with prescribed spectrum. In this work we make use of Brauer’s Theorem again, to show that most of the previous results giving sufficient conditions for the real nonnegative inverse eigenvalue problem can be derived by using Brauer’s Theorem. Moreover, the technique is constructive, and there is an algorithmic procedure to construct a matrix realizing the spectrum. In particular, we show that if either Kellogg’s realizability criterion or Borobia’s realizability criterion is satisfied, then Soto’s realizability criterion is also satisfied. None of the conv...
A connection is established between the problem of characterizing all possible real spectra of en... more A connection is established between the problem of characterizing all possible real spectra of entrywise nonnegative matrices (the so-called real nonnegative inverse eigenvalue problem) and a combinatorial process consisting in repeated application of three basic manipulations on sets of real numbers. Given realizable sets (i.e., sets which are spectra of some nonnegative matrix), each of these three elementary transformations constructs a new realizable set. This defines a special kind of realizability, called C-realizability and this is closely related to the idea of compensation. After observing that the set of all C-realizable sets is a strict subset of the set of realizable ones, we show that it strictly includes, in particular, all sets satisfying several previously known sufficient realizability conditions in the literature. Furthermore, the proofs of these conditions become much simpler when approached from this new point of view. (A. Borobia), jmoro@math.uc3m.es (J. Moro), ...
Page 1. POSITIVE MATRICES WITH PRESCRIBED SINGULAR VALUES ∗ EMEDIN MONTA NO UNIVERSIDAD DE MAGAL... more Page 1. POSITIVE MATRICES WITH PRESCRIBED SINGULAR VALUES ∗ EMEDIN MONTA NO UNIVERSIDAD DE MAGALLANES, CHILE MARIO SALAS UNIVERSIDAD CAT ´OLICA DEL NORTE, CHILE and RICARDO L ...
Page 1. ELA EXTREME SPECTRA REALIZATION BY REAL SYMMETRIC TRIDIAGONAL AND REAL SYMMETRIC ARROW MA... more Page 1. ELA EXTREME SPECTRA REALIZATION BY REAL SYMMETRIC TRIDIAGONAL AND REAL SYMMETRIC ARROW MATRICES∗ HUBERT PICKMANN, JUAN C. EGANA, AND RICARDO L. SOTO§ Abstract. We consider ...
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