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[EN] Let A be an irreducible totally nonnegative matrix with rank r and principal rank p, that is, all minors of A are nonnegative, r is the size of the largest invertible square submatrix of A and p is the size of its largest invertible principal submatrix. A triple (n,r,p) is called realizable if there exists an n*n irreducible totally nonnegative matrix A with rank r and principal rank p. In this work we present a method to construct irreducible totally nonnegative matrices associated with a realizable triple (n,r,p) and with a prescribed Jordan canonical form corresponding to its zero eigenvalue.

[EN] Let A is an element of R-nxn be a totally nonpositive matrix (t.n.p.) with rank r and principal rank p, that is, every minor of A is nonpositive and p is the size of the largest invertible principal submatrix of A. We introduce that a triple (n, r, p) will be called negatively realizable if there exists a t.n.p. matrix A of order n and such that its rank is r and its principal rank is p. In this work we extend the results obtained for irreducible totally nonnegative matrices given in Canto and Urbano (Linear Algebra Appl 551:125-146. https://doi.org/10.1016/j.laa.2018.03.045, 2018) to t.n.p. matrices. For that, we consider the sequence of the first p-indices of A and study the linear dependence relations between their rows and columns. These relations allow us to construct t.n.p. matrices associated with a triple (n, r, p) negatively realizable and a specific sequence of the first p-indices.

[EN] Let A¿Rn×n be an irreducible totally nonnegative matrix (ITN), that is, A is irreducible with all its minors nonnegative. A triple (n,r,p) is called realizable if there exists an ITN matrix A¿Rn×n with rank(A)=r and p-rank(A)=p (recall that p-rank(A) is the size of the largest invertible principal submatrix of A). Each ITN matrix associated with a realizable triple (n,r,p) has positive and distinct eigenvalues, and for the zero eigenvalue it is verified that n¿r and n¿p are the geometric and the algebraic multiplicity, respectively. Moreover, since rank(A)=r, has n¿r zero Jordan blocks whose sizes are given by the Segre characteristic, S= (s1, s2,.. . , s_n-r ), with s_i¿ p, i = 1, 2, . . . , n¿r. We know the number of zero Jordan canonical forms of ITN matrices associated with a realizable triple (n,r,p) and all these zero Jordan canonical forms. The following important question that we present in this talk deals with how to construct an ITN matrix associated with (n,r,p) and exactly with one of these Segre characteristic S corresponding to the zero eigenvalue.

[EN] Let be an irreducible totally non-negative matrix with rank r and principal rank p, that is, every minor of A is non-negative and p is the size of the largest invertible principal submatrix of A. Using Number Theory, we calculate the number of Jordan canonical forms of irreducible totally non-negative matrices associated with a realizable triple . Moreover, by using full rank factorizations of A and applying the Flanders theorem we obtain all these Jordan canonical forms. Finally, some algorithms associated with these results are given

[EN] In this paper we present a Silverman-Ho algorithm-based method to obtain a realization of a polynomial matrix. This method provides the final formulation of a minimal realization directly from a full rank factorization of a specific given matrix. Also, some classical problems in control theory such as model reduction in singular systems or the positive realization problem in standard systems are solved with this method.

[ES] En este trabajo presentamos las competencias matemáticas reflejadas en el Programa Internacional de Evaluación de Estudiantes (PISA) de la Organización para la Cooperación y Desarrollo Económico (OCDE). El objetivo principal de este programa es evaluar los sistemas de educación de todo el mundo con el fin de proporcionar información a los países, que les permita tomar decisiones para mejorar los niveles educativos y para preparar a los jóvenes para vivir en la sociedad actual. Por otra parte, la Universitat Politècnica de València (UPV) ha puesto en marcha un programa para que sus estudiantes adquieran unas competencias transversales mientras desarrollan sus estudios en la Universidad y su objetivo es preparar a los estudiantes para el mundo laboral. Los estudiantes que empiezan sus estudios de grado en la UPV tienen asignaturas de formación básica que incluyen la materia Matemáticas. Por ello, pretendemos relacionar las competencias matemáticas adquiridas en los sistemas educativos anteriores, con las competencias transversales de la UPV aplicadas a la materia Matemáticas, con el fin de conocer la similitud entre ambas y su acercamiento para facilitar la adquisición por parte de los alumnos a lo largo de su formación académica, [EN] In this paper we present the mathematical competences reflected in the Programme for International Student Assessment (PISA) of the Organization for Economic Cooperation and Development (OECD). The main objective of this program is to evaluate education systems worldwide in order to give information to the countries that allows them to make decisions to improve educational levels and to prepare young people to live in today's society. On the other hand, the Universitat Politècnica de València (UPV) has introduced a program for students to develop transversal competences while they are studying at the University, and its objective is to prepare students for the world work. Students who start their degree studies at the UPV have basic subjects that include Mathematics. For this reason we want to relate the mathematical competences acquired in the previous educational systems to the transversal competences of the UPV applied to Mathematics, in order to know the similarity between both and their approach to facilitate the acquisition by the students through their academic formation

[EN] Let A is an element of R-nxn be a totally nonnegative matrix with principal rank p, that is, every minor of A is nonnegative and p is the size of the largest invertible principal submatrix of A. We introduce the sequence of the first p-indices of A as the first initial row and column indices of a p x p invertible principal submatrix of A with rank p. Then, we study the linear dependence relations between the rows and columns indexed by the sequence of the first p-indices of A and the remaining of its rows and columns. These relations, together with the irreducibility property of some submatrices of A, allow us to present an algorithm that calculates the maximum rank of A as a function of the distribution of the first p-indices. Finally, we present a method to construct n x n totally nonnegative matrices with given rank r, principal rank p and a specific sequence of the first p-indices. (C) 2018 Elsevier Inc. All rights reserved.

[EN] In this paper we study the stability property of positive linear continuous-time systems. This property is useful to study the asymptotic behavior of a dynamical system and specifically, in positive systems. Stabilization of linear systems using feedbacks has been deeply studied during the last decades. Motivated by some results, in this paper we find conditions on the system such that the eigenvalues of the closed loop system are in the open left half plane of the complex plane C. We do this by applying a Brauer s theorem.

The final publication is available at Springer via http://dx.doi.org/10.1007/s40324-016-0073-1, [EN] A nonpositive real matrix $A= (a_{ij})_{1 \leq i, j \leq n}$ is said to be totally nonpositive (negative) if all its minors are nonpositive (negative) and it is abbreviated as t.n.p. (t.n.). In this work a bidiagonal factorization of a nonsingular t.n.p. matrix $A$ is computed and it is stored in an matrix represented by $\mathcal{BD}_{(t.n.p.)}(A)$ when $a_{11}< 0$ (or $\mathcal{BD}_{(zero)}(A)$ when $a_{11}= 0$). As a converse result, an efficient algorithm to know if an matrix $\mathcal{BD}_{(t.n.p.)}(A)$ ($\mathcal{BD}_{(zero)}(A)$) is the bidiagonal factorization of a t.n.p. matrix with $a_{11}<0$ ($a_{11}= 0$) is given. Similar results are obtained for t.n. matrices using the matrix $\mathcal{BD}_{(t.n.)}(A)$, and these characterizations are extended to rectangular t.n.p. (t.n.) matrices. Finally, the bidiagonal factorization of the inverse of a nonsingular t.n.p. (t.n.) matrix $A$ is directly obtained from $\mathcal{BD}_{(t.n.p.)}(A)$ ($\mathcal{BD}_{(t.n.)}(A)$).

In this work a technique to improve the condition number si of a simple eigenvalue lambda(i) of a matrix A is an element of C-nxn is given. This technique obtains a rank one updated matrix that is similar to A with the eigenvalue condition number of lambda(i) equal to one. More precisely, the similar updated matrix A + v(i)q*, where Av(i) = lambda(i)v(i) and q is a fixed vector, has s(i) = 1 and the remaining condition numbers are at most equal to the corresponding initial condition numbers. Moreover an expression to compute the vector q, using only the eigenvalue lambda(i) and its eigenvector v(i), is given. (C) 2016 Elsevier Ltd. All rights reserved.

[EN] Let A be a rank deficient square matrix. We characterize the unique full rank Cholesky factorization LL^T of A where the factor L is a lower echelon matrix with positive leading entries. We compute an extended decomposition for the normal matrix B^TB where B is a rectangular rank deficient matrix. This decomposition is obtained without interchange of rows and without computing all entries of the normal matrix. Algorithms to compute both factorizations are given.

[ES] Se dice que una matriz es totalmente positiva (se denota por TP) si el determinante de cada una de sus submatrices cuadradas es mayor o igual que cero, y es estrictamente totalmente positiva (denotada por STP) si todos los determinantes de sus submatrices cuadradas son positivos. En este trabajo vamos a considerar la teoría de la total positividad con considerables consecuencias y aplicaciones en diferentes áreas, como son en Biología, Física, Economía, etc. y también en la propia ciencia matemática.

Let A = (a(ij)) is an element of R-nxm be a totally nonpositive matrix with rank(A) = r <= min{n, m} and a(11) = 0. In this paper we obtain a characterization in terms of the full rank factorization in quasi-LDU form, that is, A = (L) over tilde DU where (L) over tilde is an element of R-nxr is a block lower echelon matrix, U is an element of R-rxm is a unit upper echelon totally positive matrix and D is an element of R-rxr is a diagonal matrix, with rank((L) over tilde) = rank(U) = rank(D) = r. We use this quasi-LDU decomposition to construct the quasi-bidiagonal factorization of A. Moreover, some properties about these matrices are studied. (C) 2013 Elsevier Inc. All rights reserved.

La finalidad de esta comunicación es presentar nuestro trabajo a la comunidad universitaria. Nuestra línea de investigación general se basa en estudiar la teoría matemática de los sistemas de control, concretamente en analizar las propiedades de los sistemas positivos y cuasi-positivos que se utilizan como herramientas de gestión de explotaciones agrícolas y cinegéticas, respectivamente. Para ello, ha sido necesario profundizar en la teoría matricial subyacente, en la que aparece la estructura de Jordan y la descomposición en valores singulares de matrices no invertibles, así como las factorizaciones de matrices.

Let A = (a(ij)) is an element of R-nxn be a nonsingular totally nonpositive matrix. In this paper we describe some properties of these matrices when a(11) = 0 and obtain a characterization in terms of the quasi-LDU factorization of A, where L is a block lower triangular matrix, D is a diagonal matrix and U is a unit upper triangular matrix. (c) 2012 Elsevier Inc. All rights reserved.

Given a square matrix A, a Brauer's theorem [Brauer A., Limits for the characteristic roots of a matrix. IV. Applications to stochastic matrices, Duke Math. J., 1952, 19(1), 75-91] shows how to modify one single eigenvalue of A via a rank-one perturbation without changing any of the remaining eigenvalues. Older and newer results can be considered in the framework of the above theorem. In this paper, we present its application to stabilization of control systems, including the case when the system is noncontrollable. Other applications presented are related to the Jordan form of A and Wielandt's and Hotelling's deflations. An extension of the aforementioned Brauer's result, Rado's theorem, shows how to modify r eigenvalues of A at the same time via a rank-r perturbation without changing any of the remaining eigenvalues. The same results considered by blocks can be put into the block version framework of the above theorem. © 2012 Versita Warsaw and Springer-Verlag Wien.

The main objective of this work is to model the dynamics of cynegetic populations by means of a linear control system. We want to estimate the annual corrective measures, which may be improvements or hunting, that must be implemented to achieve and maintain a study population around the carrying capacity of a certain area. Moreover, the model must be able to distinguish between different age-sex classes and lead the population to a desired distribution of individuals among these stage groups.

En este trabajo se estudian los sistemas singulares lineales de control a partir de las propiedades obtenidas para los sistemas estándares. Se obtienen realizaciones positivas de ciertas matrices de trasferencia con polos reales, analizando las condiciones para que la dimensión de la realización positiva sea minimal.

[EN] B. N. Datta and K. Datta have proposed an efficient parallel algorithm for the single-input pole assignment problem when the spectrum set to be assigned is pairwise distinct and disjoint from the spectrum of the dynamical matrix. This paper first presents a theoretical analysis of that algorithm and then presents the necessary modifications on it for an arbitrary spectrum. Based on the modified algorithm, the authors present two new algorithms for the problem and give some results on controllability that are of independent interests.