Linear Algebra and its Applications 483 (2015) 139–157

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Linear Algebra and its Applications

www.elsevier.com/locate/laa

Nonnegative persymmetric matrices with ✩ prescribed elementary divisors

Ricardo L. Soto ∗, Ana I. Julio, Mario Salas

Departamento de Matemáticas, Universidad Católica del Norte, Casilla 1280, Antofagasta, Chile a r t i c l e i n f o a b s t r a c t

Article history: The nonnegative inverse elementary divisors problem Received 11 February 2015 (NIEDP)is the problem of finding conditions for the existence Accepted 29 May 2015 of an n ×n entrywise nonnegative A with prescribed el- Available online 10 June 2015 ementary divisors. We consider the case in which the solution Submitted by M. Tsatsomeros matrix A is required to be persymmetric. Persymmetric ma- MSC: trices are common in physical sciences and engineering. They 15A18 arise, for instance, in the control of mechanical and electric vibrations. In this paper, we solve the NIEDP for n ×n matri- Keywords: ces assuming that (i)there exists a partition of the given list Persymmetric matrices Λ = {λ1, ..., λn} in sublists Λk, along with suitably chosen Companion matrices Perron eigenvalues, which are realizable by nonnegative ma- Nonnegative inverse elementary trices Ak with certain of the prescribed elementary divisors, divisors problem and (ii)a nonnegative persymmetric matrix exists with diag- onal entries being the Perron eigenvalues of the matrices Ak, with certain of the prescribed elementary divisors. Our results generate an algorithmic procedure to compute the structured solution matrix. © 2015 Elsevier Inc. All rights reserved.

✩ Supported by FONDECYT 1120180, Chile, CONICYT-PCHA/Doc.Nac./2013-63130010, Chile. * Corresponding author. E-mail addresses: [email protected] (R.L. Soto), [email protected] (A.I. Julio), [email protected] (M. Salas). http://dx.doi.org/10.1016/j.laa.2015.05.032 0024-3795/© 2015 Elsevier Inc. All rights reserved. 140 R.L. Soto et al. / Linear Algebra and its Applications 483 (2015) 139–157

1. Introduction

Let A ∈ Cn×n and let ⎡ ⎤ J ⎢ n1(λ1) ⎥ ⎢ J ⎥ −1 ⎢ n2(λ2) ⎥ J(A)=S AS = ⎢ . ⎥ ⎣ .. ⎦

Jnk(λk) be the Jordan canonical form of A (hereafter JCF of A). The ni × ni submatrices ⎡ ⎤ λi 1 ⎢ . ⎥ ⎢ λ .. ⎥ J (λ )=⎢ i ⎥ ,i=1, 2,...,k ni i ⎢ . ⎥ ⎣ .. 1 ⎦

λi are called the Jordan blocks of J(A). The elementary divisors of A are the polynomials − ni (λ λi) , that is, the characteristic polynomials of Jni (λi), i =1, ..., k. The nonnegative inverse elementary divisors problem (hereafter NIEDP) is the problem of determining n1 n2 necessary and sufficient conditions under which the polynomials (λ −λ1) , (λ −λ2) , ..., n (λ −λk) k , n1 +···+nk = n, are the elementary divisors of an n ×n A [5,6,12]. In [5,6] Minc considers the NIEDP modulo the nonnegative inverse eigenvalue problem (hereafter NIEP), which is the problem of finding necessary and sufficient conditions for the existence of a nonnegative matrix with prescribed spectrum. We approach the NIEDP in the same way. In particular, Minc showed that if A is a diagonalizable positive matrix (diagonalizable positive doubly ), then there exists a positive matrix B (positive B) with the same spectrum as A, and with any prescribed elementary divisors. In [12,15], the authors completely solve the NIEDP for lists of real numbers Λ = {λ1, ..., λn} satisfying: i) λ1 >λ2 ≥ ··· ≥ λn ≥ 0, and ii) λ1 > 0 >λ2 ≥ ··· ≥ λn, and for lists of complex numbers satisfying iii)Reλi < 0, |Re λi|≥|Im λi|, i =2, ..., n. Sufficient conditions are also given in [12,15] for the general case. n A matrix A =(aij)i,j=1 is said to have constant row sums if all its rows add up n to the same constant γ, i.e. aij = γ, i =1, ..., n. The set of all matrices with j=1 T constant row sums equal to γ is denoted by CSγ . It is clear that e =(1, 1, ..., 1) is an eigenvector of any matrix A ∈CSγ , corresponding to the eigenvalue γ. A nonnegative T matrix A is called stochastic if A ∈CS1 and is called doubly stochastic if A, A ∈CS1. ∈CS T ∈CS Amatrix A λ1 or A, A λ1 , is called generalized stochastic or generalized doubly stochastic, respectively. The relevance of the real matrices with constant row sums is due Download English Version: https://daneshyari.com/en/article/4598933

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