A Family of Nonnegative Matrices with Prescribed Spectrum and Elementary Divisors 1

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1 International Mathematical Forum, Vol, 06, no 3, HIKARI Ltd, wwwm-hikaricom A Family of Nonnegative Matrices with Prescribed Spectrum and Elementary Divisors Ricardo L Soto Dpto Matemáticas, Universidad Católica del Norte 70709, Antofagasta, Chile Elvis Valero Dpto Matemáticas, Universidad de Tarapacá Arica, Chile Copyright c 06 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 eigenvalues 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 Both results have been exploited in connection with the nonnegative inverse eigenvalue 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 elementary divisors, generalizing previous results We also apply our results to the nonnegative pole assigment problem Mathematics Subject Classification: 5A8, 5A9 Keywords: nonnegative inverse eigenvalue problem, nonnegative inverse elementary divisors problem Work supported by Fondecyt 080 Corresponding author

2 600 Ricardo L Soto and Elvis Valero Introduction The origin of the present paper is a perturbation result due to Brauer [], 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 [0], giving rise to a number of realizability criteria (see [] and the references therein) Closer to our approach in this paper is Rado s extension of Brauer s result, Theorem below, which was first used by Perfect in [9] and later in [] 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 [, ], as well as for the nonnegative inverse elementary divisors problem (NIEDP) [3, 5, 6] In this paper we use the Rado result (Theorem below) in a more general context, to construct in Section, a family of nonnegative matrices with prescribed spectrum In Section 3 we construct a family of nonnegative matrices with prescribed elementary divisors In Section we apply our results to the pole assigment problem We denote the spectrum of a matrix A as σ(a) Rank-r perturbations The following result, due to R Rado and introduced by Perfect [9] in 955, 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 (Rado) Assume that for r n all following are given: i) let A C n n be an arbitrary matrix with σ(a) = {λ,, λ n }; ii) let U = [u u r ] C n r be a matrix of rank r such that Au i = λ i u i for each i =,, r; iii) let V = [v v r ] C n r be an arbitrary matrix; iv) let Ω = diag(λ,, λ r ) C r r ; Then σ(a + UV T ) = σ(ω + V T U) {λ r+,, λ n } We state an interesting consequence of Theorem Corollary Assume that for r n all following are given: i) let A C n n be an arbitrary matrix with σ(a) = {λ,, λ n }; ii) let U = [u u r ] C n r be a matrix of rank r such that Au i = λ i u i for each i =,, r; iii) let W = [w w r ] C n r be an arbitrary matrix with W T U = I r ;

3 A family of nonnegative matrices with prescribed spectrum and 60 iv) let C C r r be an arbitrary matrix and let Ω = diag(λ,, λ r ) C r r ; Then σ(a + UCW T ) = σ(ω + C) {λ r+,, λ n } Proof If we take V = W C T A + UV T = A + UCW T in Theorem, then Ω + V T U = Ω + C and The advantage of this result with respect to Theorem is that Corollary permits to construct a family of matrices with a prescribed spectrum since the set of eigenvalues {µ, µ,, µ r } of Ω + C does not depend on the matrix W : namely, for any W such that W T U = I r we can construct a new matrix A + UCW T with the desired spectrum {µ,, µ r, λ r+,, λ n } Corollary Assume all the following are given: i) let S = [s ij ] r i,j= C r r be an arbitrary matrix with σ(s) = {ρ,, ρ r }; ii) for each i =,, r let A i C n i n i be an arbitrary matrix with an eigenvalue s ii ; iii) for each i =,, r let u i C n i be an eigenvector of A i associated with the eigenvalue s ii ; iv) for each i =,, r let w i C n i be a vector such that wi T u i = Then the matrix A s u w T s r u wr T s u w T A s r u w T r () s r u r w T s r u r w T A r has spectrum {σ(a ) {s }} {σ(a r ) {s rr }} {ρ,, ρ r } Proof We are going to apply Corollary Therefore: Let A A 0 A = C(n ++n r) (n ++n r) ; 0 0 A r Let u u 0 U = C(n ++n r) r 0 0 u r and observe that for each i =,, r, the i th column of U is an eigenvector of A with eigenvalue s ii ;

4 60 Ricardo L Soto and Elvis Valero Let W = w w 0 C(n ++n r) r 0 0 w r and observe that W T U = I r Let C = 0 s s r s 0 s r s r s r 0 Then A+UCW T, as given in (), has spectrum {ρ,, ρ r } {σ(a) {s,, s rr }} Remark Assume in Corollary that the matrices A i C n i n i are all nonnegative with constant row sums equal to s ii (the Perron eigenvalue) Then u i = e = (,,, ) T, i =,, r Let C an r r nonnegative matrix Moreover, assume that ω i is nonnegative, i =,, r The matrix A+UCW T in () is nonnegative with the spectrum {σ(a ) {s }} {σ(a r ) {s rr }} {ρ,, ρ r } Hence, under these conditions we may construct a family of nonnegative matrices with a prescribed spectrum Example Let Λ = {6, 3, 3, 5, 5} We look for a family of nonnegative matrices with spectrum Λ Then we take the partition Λ = {6, 5}, Λ = {3, 5}, Λ 3 = {3} with Γ = {5 5}, Γ = {5 5}, Γ 3 = {} being realizable by Then the matrix A = A = A A = A [ A 3 = ], A 3 = [] has eigenvalue 5, 5,, 5, 5 Moreover

5 A family of nonnegative matrices with prescribed spectrum and 603 U = , W = α γ η α γ η β δ θ β δ θ 0 0 with W T U = I Next we compute 5 0 B = 5 0, with C = B diagb = 0, B has spectrum {6, 3, 3} and diagonal entries {5, 5, } Thus, for η = θ = γ = β = 0; 0 α, δ, the matrix η 5 η θ θ 5 + η η θ θ A + UCW T = α α + β β α α + β + 5 β 0 γ γ δ δ + gives a family of nonnegative matrices with spectrum Λ = {6, 3, 3 5, 5} We can also obtain a family of persymmetric nonnegative matrices with prescribed 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 symmetric and persymmetric, play an important role As the superscript T, in A T, denotes the transpose of A, the superscript F, in A F, denotes the fliptranspose of A, which flips A across its skew-diagonal If A = (a ij ) mn, then A F = (a n j+,m i+ ) nm A matrix A is said to be persymmetric if A F = A, that is, if it is symmetric across its lower-left to upper-right diagonal In [], the authors give a persymmetric version of Theorem, which we use in the following Corollary 3 Assume that for r n all following are given: i) let A C n n be a persymmetric nonnegative matrix with σ(a) = {λ,, λ n }; ii) let U = [u u r ] C n r be a matrix of rank r such that Au i = λ i u i for each i =,, r; iii) let C C r r be persymmetric nonnegative matrix and let Ω = diag(λ,, λ r ) C r r ; Then A + UCU F is persymmetric nonnegative matrix with spectrum σ(a + UCU F ) = {µ,, µ r } {λ r+,, λ n }, where µ,, µ r are the eigenvalues of Ω + CU F U (Ω + C if the columns of U are taken as orthonormal)

6 60 Ricardo L Soto and Elvis Valero Proof It is clear that UCU F is persymmetric nonnegative and then A + UCU F es also persymmetric nonnegative From Theorem and Remark, A + UCU F has the spectrum σ(a) Example We construct a family of persymmetric nonnegative matrices with spectrum Λ = {6,,,, 3, 3} We take the partition Λ = Λ Λ with Λ = {6,, 3}, Λ = {,, 3} and the associated realizable lists Γ = {5, 3}, Γ = {5,, 3} Let [ A 0 A = 0 A F ] = , where A and A F realizan Γ and Γ respectively Let U be the matrix of the normalized eigenvectors of A corresponding to the eigenvalues 5, 5 : U = and let C = [ 0 y x 0 ] Then for all x, y 0, A + UCU F = y 5 y 5 y y 5 y y y 5 y y 5 7 x 5 5 x 5 x x 5 5 x 5 x x x 5 x is persymmetric nonnegative In particular for x = y =, A + UCU F has the spectrum Λ

7 A family of nonnegative matrices with prescribed spectrum and Inverse elementary divisors problem In this section we show how to construct a family of nonnegative matrices with prescribed elementary divisors Here we exploit Corollary for the nonnegative case (as in Remark ) Let A C n n and let J(A) = S AS = J n (λ ) J n (λ ) J nk(λk) be the Jordan canonical form of A The n i n i submatrices λ i λ J ni (λ i ) = i, i =,,, k λ i are the Jordan blocks of J(A) The elementary divisors of A are the polynomials (λ λ i ) n i, that is, the characteristic polynomials of J ni (λ i ), i =,, k The nonnegative inverse elementary divisor problem (NIEDP) is the problem of determining necessary and sufficient conditions under which the polynomials (λ λ ) n, (λ λ ) n,, (λ λ k ) n k, n + + n k = n, are the elementary divisors of an n n nonnegative matrix A [5, 6, 7] The NIEDP contains the NIEP and both problems remain unsolved (they have been solved only for n ) The following result gives a sufficient condition for the existence and construction of a nonnegative matrix with prescribed spectrum and elementary divisors Corollary 3 Let Λ = {λ, λ,, λ n } be a list of complex numbers, with Λ = Λ, λ max i λ i, i =,, n, and n λ i 0 Assume that there exists a partition Λ = Λ 0 Λ Λ r, where some of the lists Λ i, i =,, r, can be empty, such that: i) let S = [s ij ] r i,j= is an r r nonnegative matrix with spectrum σ(s) = Λ 0 = {λ,, λ r }; ii) for each i =,, r, there exists a (p i + ) (p i + ) nonnegative matrix A i, with constant row sums equal to s ii, spectrum Γ i = {s ii, λ i, λ i,, λ ipi }, and prescribed elementary divisors associated with Γ i i=

8 606 Ricardo L Soto and Elvis Valero iii) for each i =,, r let u i = e be an eigenvector of A i corresponding to the eigenvalue s ii iv) Let A be the block diagonal matrix A = diag{a,, A r } and let U = [û û r ] such that Aû i = s ii û i v) for each i =,, r let w i be a nonnegative vector such that w T i ûi =, and let W = [w w r ] Then the nonnegative matrix A+UCW T, where C = S diag{s, s,, s rr }, has spectrum Λ and the prescribed elementary divisors associated with Λ i, i =,, r Proof From ii) iv) A = diag{a,, A r } is nonnegative with spectrum Γ Γ r and Aû i = s ii û i, i =,, r Then from i) and Remark the result follows Example 3 Let Λ = {7,,,, +3i, 3i} We want to construct a family of nonnegative matrices with spectrum Λ, and with elementary divisors (λ 7), (λ ), (λ + ), λ + λ + 0 Then we take the partition Λ 0 = {7, + 3i, 3i}, Λ = {, }, Λ = {}, Λ 3 = with Γ = {,, }, Γ = {, }, Γ 3 = {0} We compute the nonnegative matrix 0 3 S = 3 8 with spectrum Λ and diagonal entries,, 0, and the realizing matrices 0 0 A = 7 + [ ] = 6 0 [ ] 0 A = and A 0 3 = [0], with spectra Γ, Γ, Γ 3, respectively Observe that the matrix A has elementary divisors (λ ), (λ + ) Let α γ η 0 0 α γ η 0 0 W = β δ θ and U = β δ θ ,

9 A family of nonnegative matrices with prescribed spectrum and 607 Then W T U = I, and A A = A = A 3 + UCW T 3η 3η 3θ 3θ 3 3η + 3 3η 3θ 3θ 3 3η + 3η 0 3θ 3θ 3 α + 8η γ 7γ 0 7δ 7 7δ is a family of matrices with spectrum Λ and the desired elementary divisors In particular for η = γ = θ = β = 0, and 0 α, δ we have the nonnegative family B = α 3 3α 0 0 8, α 3 3α δ 7 7δ 0 with spectrum Λ and elementary divisors (λ 7), (λ ), (λ + ), λ + λ + 0 We can also solve the inverse elementary divisors problem for a family of nonnegative matrices with constant row sums and spectrum Λ = {λ,, λ n } in the following cases, which have been solved for a single nonnegative matrix in [3, 5, 3]: i) λ > λ λ n 0 ii) λ > 0 λ λ n iii) Reλ i 0, Reλ i Imλ i iv) Reλ i 0, 3Reλ i Imλi For each one of these cases, we may apply the same techniques used in [3, 5, 3] for a single nonnegative matrix For example, in the first case i) Λ = {5, 3, 3, 3}, we use from [8], the Perfect matrix P = 0 0 0,

10 608 Ricardo L Soto and Elvis Valero to obtain a positive matrix A = P DP = 3 3 7, where D = diag{5, 3, 3, 3} If E i,j is the matrix with in position (i, j) and zeros elsewhere, then M = A + ap E 3, P = a + 3 a a + 3 a a a + 7 with 0 a, is a nonnegative matrix with constant row sums equal to 5, and with elementary divisors J (5), J (3), J (3) For J (5), J 3 (3) we have with E = ae 3, + be,3, M = A + ap EP = a + b + 3 a + b + b a + b a + 3 a + b + b a + b b, b b 7 b b + which for 0 a, b, is nonnegative with the desired JCF For the second case ii) λ > 0 λ λ n, we apply Brauer result: M = = a a 0 a 0 a a a a a + 0 +,, [ ] 0 which, for 0 a, is nonnegative, with constant row sums λ = 0, and elementary divisors J (0), J ( ), J ( ) In the same way, for the list Λ = {, + i, i, + i i}

11 A family of nonnegative matrices with prescribed spectrum and 609 we have M = = a a a 0 a a a a 0 a + 3 a 0 a +, 0 a, [ ] 3 a a with JCF J(M) = i i i i Nonnegative pole assigment problem In [] the authors show applications of Theorem (Rado Theorem), to deflation techniques and the pole assigment problem for multi-imput multi-output (MIMO) systems defined by pairs of matrices (A, B) Corollary may also be applied to deflation techniques and the pole assignment problem to obtain a nonnegative solution matrix Let A be an n n matrix, let B be an n r matrix with σ(a) = {λ, λ,, λ n } Let {µ, µ,, µ r } be a list of numbers The pole assignment problem for MIMO systems asks conditions on the pair (A, B) for the existence of a matrix F such that σ(a + BF T ) = {µ, µ,, µ r, λ r+, λ r+,, λ n } For the nonnegative pole assigment problem we have the following result: Corollary Let (A, B) be a given pair of matrices with A an n n nonnegative matrix and B an n r matrix Let σ(a) = {λ, λ,, λ n } and let µ = {µ, µ,, µ r } be a list of numbers Let X = [α x α r x r ] be an n r matrix such that rankx = r and A T x i = λ i x i, i =,,, r If there

12 60 Ricardo L Soto and Elvis Valero exists a column b j = [b j, b j,, b rj ] T of the matrix B, with all its entries being nonnegative such that b T j x i 0, i =,,, r, σ(ω + [b j b j ] T X) = {µ, µ,, µ r } and r i= α ix si 0, s =,,, n, then there exists a nonnegative matrix F such that the nonnegative matrix A + BF T has spectrum {µ,, µ r, λ r+,, λ n } Proof Let the n r matrix C T = [b j b j ] = B[e j e j ], where e j is the j th column of the identity matrix of order r Then C T X T = B[e j e j ]X T b r j i= α ix i b r j i= α ix ni = 0 b r nj i= α ix i b r nj i= α ix ni If we take F T = [e j e j ]X T then 0 A + C T X T = A + B[e j e j ]X T = A + BF T and for Theorem σ(a + BF T ) = σ(a + C T X T ) = σ(ω + [e j e j ] T B T X) {λ r+,, λ n } Example Consider the pair (A, B) and µ = {8 + 9, 8 9} where A = and B = = [b b ], with σ(a) = {6, 3, 3, 5, 5} We compute the eigenvector of A T : x λ=6 = (,,,, ) T, x λ=3 = (,,,, )T x λ= 05 = {( 5, 5, 0,, 0) T, ( 35, 79 0, 7, 0, )T }

13 A family of nonnegative matrices with prescribed spectrum and 6 Since for b = 0 0 0, bt x λ=6 0, and b T x λ=3 0, we may change the eigenvalues λ = 6 and λ = 3 to µ = and µ = 8 9, respectively To do this, let [ ] 0 0 C T = [b b ] = B = B [e e ] with [ X T α α = α α α α α α α α Since σ(ω + [b b ] T X) = {µ, µ }, must be 6 + 5α α = (6 + 5α )(3 + α ) (5α )(α ) = (8 + 9)(8 9) ] Then α =, α = Taking F T = [e e ] X T, we have that is nonnegative and A + BF T = σ(a + BF T ) = σ(ω + [e e ] T B T X) {3, 5, 5} where Ω + [e e ] T B T X = [ 5 5 ] has the spectrum 8 + 9, 8 9 References [] A Brauer, Limits for the characteristic roots of a matrix IV: Aplications to stochastic matrices, Duke Math J, 9 (95),

14 6 Ricardo L Soto and Elvis Valero [] R Bru, R Canto, R Soto, AM Urbano, A Brauer s theorem and related results, Central European Journal of Mathematics, 0 (0), [3] RC Díaz, RL Soto, Nonnegative inverse elementary divisors problem in the left half plane, Linear and Multilinear Algebra, 6 (05), [] AI Julio, RL Soto, Persymmetrix and bisymmetric nonnegative inverse eigenvalue problem, Linear Algebra and its Appl, 69 (05), [5] H Minc, Inverse elementary divisor problem for nonnegative matrices, Proceedings of the American Mathematical Society, 83 (98), no, [6] H Minc, Inverse elementary divisor problem for doubly stochastic matrices, Linear and Multilinear Algebra, (98), -3 [7] H Minc, Nonnegative Matrices, John Wiley & Sons, New York, 988 [8] H Perfect, Methods of constructing certain stochastic matrices, Duke Math J, 0 (953), [9] H Perfect, Methods of constructing certain stochastic matrices II, Duke Math J, (955), [0] RL Soto, Existence and construction of nonnegative matrices with prescribed spectrum, Linear Algebra and its Appl, 369 (003), [] RL Soto, O Rojo, Applications of a Brauer theorem in the nonnegative inverse eigenvalue problem, Linear Algebra and its Appl, 6 (006), [] RL Soto, O Rojo, J Moro, A Borobia, Symmetric nonnegative realization of spectra, Electronic J of Linear Algebra, 6 (007), -8 [3] RL Soto, J Ccapa, Nonnegative matrices with prescribed elementary divisors, Electron J of Linear Algebra, 7 (008),

15 A family of nonnegative matrices with prescribed spectrum and 63 [] RL Soto, A family of realizability criteria for the real and symmetric nonnegative inverse eigenvalue problem, Numerical Linear Algebra with Appl, 0 (0), [5] RL Soto, RC Díaz, H Nina, M Salas, Nonnegative matrices with prescribed spectrum and elementary divisors, Linear Algebra and its Appl, 39 (03), [6] RL Soto, AI Julio, M Salas, Nonnegative persymmetric matrices with prescribed elementary divisors, Linear Algebra and its Appl, 83 (05) Received: May, 06; Published: June 6, 06