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A Family of Nonnegative Matrices with Prescribed Spectrum and Elementary Divisors1

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A Family of Nonnegative Matrices with Prescribed Spectrum and Elementary Divisors1 International Mathematical Forum, Vol. 11, 2016, no. 13, 599 - 613 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2016.6448 A Family of Nonnegative Matrices with Prescribed Spectrum and Elementary Divisors1 Ricardo L. Soto2 Dpto. Matem´aticas,Universidad Cat´olicadel Norte 1270709, Antofagasta, Chile Elvis Valero Dpto. Matem´aticas,Universidad de Tarapac´a 1010069 Arica, Chile Copyright c 2016 Ricardo L. Soto and Elvis Valero. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract A perturbation result, due to Rado, shows how to modify r eigen- values of a matrix of order n, via a perturbation of rank r ≤ n, without changing any of the n − r remaining eigenvalues. This result extended a previous one, due to Brauer, on perturbations of rank 1. Both results have been exploited in connection with the nonnegative inverse eigen- value problem and the nonnegative inverse elementary divisors problem. In this paper, we use the Rado result from a more general point of view, constructing a family of matrices with prescribed spectrum and elemen- tary divisors, generalizing previous results. We also apply our results to the nonnegative pole assigment problem. Mathematics Subject Classification: 15A18, 15A29 Keywords: nonnegative inverse eigenvalue problem, nonnegative inverse elementary divisors problem 1Work supported by Fondecyt 1120180. 2Corresponding author 600 Ricardo L. Soto and Elvis Valero 1 Introduction The origin of the present paper is a perturbation result due to Brauer [1], which shows how to modify one single eigenvalue of a matrix, via a rank-one perturbation, without changing any of the remaining eigenvalues. This result was first used in connection with the nonnegative inverse eigenvalue problem (NIEP) by Perfect [8], and later in [10], giving rise to a number of realizability criteria (see [14] and the references therein). Closer to our approach in this paper is Rado's extension of Brauer's result, Theorem 2:2 below, which was first used by Perfect in [9] and later in [11] to derive sufficient conditions for the NIEP to have a solution. Both, Brauer and Rado results, have been also used to derive efficient sufficient conditions for the real and symmetric nonnegative inverse eigenvalue problem [11, 12], as well as for the nonnegative inverse elementary divisors problem (NIEDP) [3, 15, 16]. In this paper we use the Rado result (Theorem 2.1 below) in a more general context, to construct in Section 2; a family of nonnegative matrices with prescribed spectrum. In Section 3 we construct a family of nonnegative matrices with prescribed elementary divisors. In Section 4 we apply our results to the pole assigment problem. We denote the spectrum of a matrix A as σ(A): 2 Rank-r perturbations The following result, due to R. Rado and introduced by Perfect [9] in 1955, is an extension of the above mentioned Brauer result. It shows how to change r eigenvalues of an n×n matrix A via a perturbation of rank r, without changing any of the remaining n − r eigenvalues. Theorem 2.1 (Rado) Assume that for r ≤ n all following are given: n×n i) let A 2 C be an arbitrary matrix with σ(A) = fλ1; :::; λng; n×r ii) let U = [u1 j · · · j ur] 2 C be a matrix of rank r such that Aui = λiui for each i = 1; :::; r; n×r iii) let V = [v1 j · · · j vr] 2 C be an arbitrary matrix; r×r iv) let Ω = diag(λ1; :::; λr) 2 C ; T T Then σ(A + UV ) = σ(Ω + V U) [ fλr+1; :::; λng: We state an interesting consequence of Theorem 2:1 Corollary 2.1 Assume that for r ≤ n all following are given: n×n i) let A 2 C be an arbitrary matrix with σ(A) = fλ1; :::; λng; n×r ii) let U = [u1 j · · · j ur] 2 C be a matrix of rank r such that Aui = λiui for each i = 1; :::; r; n×r T iii) let W = [w1 j · · · j wr] 2 C be an arbitrary matrix with W U = Ir; A family of nonnegative matrices with prescribed spectrum and ... 601 r×r r×r iv) let C 2 C be an arbitrary matrix and let Ω = diag(λ1; :::; λr) 2 C ; T Then σ(A + UCW ) = σ(Ω + C) [ fλr+1; :::; λng: Proof. If we take V = WCT in Theorem 2:1, then Ω + V T U = Ω + C and A + UV T = A + UCW T : The advantage of this result with respect to Theorem 2:1 is that Corollary 2.1 permits to construct a family of matrices with a prescribed spectrum since the set of eigenvalues fµ1; µ2; :::; µrg of Ω + C does not depend on the matrix T W : namely, for any W such that W U = Ir we can construct a new matrix T A + UCW with the desired spectrum fµ1; :::; µr; λr+1; :::; λng. Corollary 2.2 Assume all the following are given: r r×r i) let S = [sij]i;j=1 2 C be an arbitrary matrix with σ(S) = fρ1; :::; ρrg; ni×ni ii) for each i = 1; :::; r let Ai 2 C be an arbitrary matrix with an eigenvalue sii; ni iii) for each i = 1; :::; r let ui 2 C be an eigenvector of Ai associated with the eigenvalue sii; ni T iv) for each i = 1; :::; r let wi 2 C be a vector such that wi ui = 1: Then the matrix 2 T T 3 A1 s12u1w2 ··· s1ru1wr 6 s u wT A ··· s u wT 7 6 21 2 1 2 2r 2 r 7 6 . .. 7 (1) 4 . 5 T T sr1urw1 sr2urw2 ··· Ar has spectrum fσ(A1) − fs11gg [ ::: [ fσ(Ar) − fsrrgg [ fρ1; :::; ρrg: Proof. We are going to apply Corollary 2.1. Therefore: Let 2 3 A1 0 ::: 0 6 0 A2 ::: 0 7 6 7 (n1+:::+nr)×(n1+:::+nr) A = 6 . .. 7 2 C ; 4 . 5 0 0 :::Ar Let 2 3 u1 0 ::: 0 6 0 u2 ::: 0 7 6 7 (n1+:::+nr)×r U = 6 . .. 7 2 C 4 . 5 0 0 : : : ur and observe that for each i = 1; :::; r; the ith column of U is an eigenvector of A with eigenvalue sii; 602 Ricardo L. Soto and Elvis Valero Let 2 3 w1 0 ::: 0 6 0 w2 ::: 0 7 6 7 (n1+:::+nr)×r W = 6 . .. 7 2 C 4 . 5 0 0 : : : wr T and observe that W U = Ir: Let 2 3 0 s12 : : : s1r 6 s 0 : : : s 7 6 21 2r 7 C = 6 . .. 7 : 4 . 5 sr1 sr2 ::: 0 T Then A+UCW ; as given in (1), has spectrum fρ1; :::; ρrg[fσ(A)−{s11; :::; srrgg: ni×ni Remark 2.1 Assume in Corollary 2.2 that the matrices Ai 2 C are all nonnegative with constant row sums equal to sii (the Perron eigenvalue). Then T ui = e = (1; 1; :::; 1) ; i = 1; :::; r: Let C an r×r nonnegative matrix. Moreover, T assume that !i is nonnegative, i = 1; :::; r: The matrix A+UCW in (1) is non- negative with the spectrum fσ(A1)−fs11gg[· · ·[fσ(Ar)−fsrrgg[fρ1; :::; ρrg: Hence, under these conditions we may construct a family of nonnegative ma- trices with a prescribed spectrum. Example 2.1 Let Λ = f6; 3; 3; −5; −5g: We look for a family of nonnegative matrices with spectrum Λ: Then we take the partition Λ1 = f6; −5g; Λ2 = f3; −5g; Λ3 = f3g with Γ1 = f5: − 5g; Γ2 = f5: − 5g; Γ3 = f2g being realizable by 0 5 A = A = ;A = [2] : 1 2 5 0 3 Then the matrix 2 0 5 0 0 0 3 2 3 A1 6 5 0 0 0 0 7 6 7 A = 4 A2 5 = 6 0 0 0 5 0 7 6 7 A3 4 0 0 5 0 0 5 0 0 0 0 2 has eigenvalue 5; 5; 2; −5; −5: Moreover A family of nonnegative matrices with prescribed spectrum and ... 603 2 1 0 0 3 2 α γ η 3 6 1 0 0 7 6 1 − α −γ −η 7 6 7 6 7 U = 6 0 1 0 7 ;W = 6 β δ θ 7 , 6 7 6 7 4 0 1 0 5 4 −β 1 − δ −θ 5 0 0 1 0 0 1 with W T U = I: Next we compute 2 5 0 1 3 2 0 0 1 3 B = 4 1 5 0 5 ; with C = B − diagB = 4 1 0 0 5 : 0 4 2 0 4 0 B has spectrum f6; 3; 3g and diagonal entries f5; 5; 2g: Thus, for η = θ = γ = β = 0; 0 ≤ α; δ ≤ 1; the matrix 2 η 5 − η θ −θ 1 3 6 5 + η −η θ −θ 1 7 T 6 7 A + UCW = 6 α −α + 1 β −β + 5 0 7 6 7 4 α −α + 1 β + 5 −β 0 5 4γ −4γ 4δ −4δ + 4 2 gives a family of nonnegative matrices with spectrum Λ = f6; 3; 3 − 5; −5g: We can also obtain a family of persymmetric nonnegative matrices with pre- scribed spectrum. Persymmetric matrices are common in physical sciences and engineering. They arise, for instance, in the control of mechanical and electric vibrations, where the eigenvalues of the Gram matrix, which is sym- metric and persymmetric, play an important role. As the superscript T; in AT ; denotes the transpose of A; the superscript F; in AF ; denotes the flip- transpose of A; which flips A across its skew-diagonal.
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