Spec. Matrices 2019; 7:230–245
Research Article Open Access Special Issue Dedicated to Charles R. Johnson
Macarena Collao, Mario Salas, and Ricardo L. Soto* Toeplitz nonnegative realization of spectra via companion matrices https://doi.org/10.1515/spma-2019-0017 Received July 25, 2019; accepted November 26, 2019
Abstract: The nonnegative inverse eigenvalue problem (NIEP) is the problem of nding conditions for the exis- tence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = {λ1, . . . , λn}. If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper we consider the NIEP for a Toeplitz realizing matrix A, and as far as we know, this is the rst work which addresses the Toeplitz nonnegative realization of spectra. We show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. We note that, as a consequence, a realizable list Λ = {λ1, . . . , λn} of complex numbers in the left-half plane, that is, with Re λi ≤ 0, i = 2, . . . , n, is in particular realizable by a Toeplitz matrix. Moreover, we show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices. We also propose a Matlab Toeplitz routine to compute a Toeplitz solution matrix.
Keywords: Toeplitz nonnegative inverse eigenvalue problem, Unit Hessenberg Toeplitz matrix, Symmetric nonnegative block Toeplitz matrix, Universal realizability
MSC: 15A29, 15A18 Dedicated with admiration and special thanks to Charles R. Johnson.
1 Introduction
The nonnegative inverse eigenvalue problem (hereafter NIEP) is the problem of nding necessary and su cient conditions for a list Λ = {λ1, λ2, . . . , λn} of complex numbers to be the spectrum of an n × n entrywise nonnegative matrix . If there exists a nonnegative matrix A with spectrum Λ, we say that Λ is realizable and that A is a realizing matrix. The NIEP has been completely solved for n = 3 [8] and for n = 4 [10], and n X independently in [21]. For n ≥ 5 the NIEP remains unsolved, although for n = 5 with λi = 0, the problem i=1 was solved in [6]. In this paper we consider the NIEP for Toeplitz realizing matrices. An n × n Toeplitz matrix
Macarena Collao: Departamento de Matemáticas, Universidad Católica del Norte, Casilla 1280, Antofagasta, Chile, E-mail: [email protected] Mario Salas: Departamento de Matemáticas, Universidad Católica del Norte, Casilla 1280, Antofagasta, Chile, E-mail: [email protected] *Corresponding Author: Ricardo L. Soto: Departamento de Matemáticas, Universidad Católica del Norte, Casilla 1280, Antofagasta, Chile, E-mail: [email protected]
Open Access. © 2020 Macarena Collao et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. Toeplitz nonnegative realization of spectra via companion matrices Ë 231 is a matrix of the form t0 t1 t2 ··· tn−1 . . t t t .. . −1 0 1 . . T = . . . t−2 t−1 . . t2 ...... . . . . t1 t−n+1 ··· t−2 t−1 t0
T is circulant if t−i = tn−i , i = 1, 2, ... , n − 1.
It is well known that the inverse eigenvalue problem (IEP) is always solvable for Toeplitz matrices: a Toeplitz −1 circulant matrix T = F DF can always be obtained with complex spectrum Λ = {λ1, ... , λn}, where F is the matrix of the discrete Fourier transform and D = diag{λ1, ... , λn}. In contrast to this result, we are interested in the nonnegativity property of a realizing matrix. In this work we show, under certain conditions, that a list of complex numbers is the spectrum of a Toeplitz nonnegative matrix, which is not necessarily circulant. In [9] the authors prove that an n×n complex nonderogatory matrix is similar to a unique unit upper Hessenberg Toeplitz matrix, and that every n × n complex matrix with n ≤ 4 is similar to a Toeplitz matrix. As far as we know, this paper is the rst work which addresses the nonnegative Toeplitz realization of spectra. Although Toeplitz matrices are persymmetric, the NIEP for persymmetric matrices (see [5, 19]) and the NIEP for Toeplitz matrices are dierent. For example, the list Λ = {6, 5, 1} is the spectrum of the nonnegative persymmetric matrix 3 0 2 P = 0 6 0 . 2 0 3 However, it is easy to check that Λ cannot be the spectrum of a nonnegative matrix with diagonal entries 4, 4, 4. In this paper we give su cient conditions for the existence and construction of a nonnegative Toeplitz matrix with prescribed spectrum.
n α A real matrix A = aij i,j=1 is said to have constant row sums if all its rows sum up to the same constant , n X i.e., aij = α, i = 1, ... , n. j=1 The set of all real matrices with constant row sums equal to α is denoted by CSα. It is clear that e = T [1, 1, ... , 1] is an eigenvector of any matrix in CSα, corresponding to the eigenvalue α. The relevance of matrices with constant row sums is due to the well known fact that if λ1 is the desired Perron eigenvalue, then the problem of nding a nonnegative matrix with spectrum Λ = {λ1, ... , λn} is equivalent to the prob- lem of nding a nonnegative matrix in CSλ1 with spectrum Λ.
The paper is organized as follows: In Section 2, we show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. As a consequence, a realizable list of complex numbers in the left-half plane, that is, a list Λ = {λ1, ... , λn} with Re λi ≤ 0, i = 2, ... , n, which includes real and complex lists of Suleimanova [20] and Šmigoc [14] type, is in particular realizable by a Toeplitz matrix. In Section 3, we give su cient conditions for the existence of a symmetric nonnegative block Toeplitz matrix with prescribed Jordan canon- ical form. We also give examples to illustrate the results. Finally, we implement a Matlab Toeplitz routine for computing the desired Toeplitz matrix, which is introduced in the Appendix, Section 4.
2 Companion matrices are similar to Toeplitz ones
In this Section, we show that an n × n nonnegative companion matrix C is similar to a nonnegative Toeplitz matrix T by a lower triangular similarity. This result, interesting in itself, gives rise to a su cient condition for 232 Ë Macarena Collao, Mario Salas, and Ricardo L. Soto the existence and construction of a nonnegative Toeplitz matrix with prescribed complex spectrum. It allows us to show that a list Λ = {λ1, ... , λn} of complex numbers in the left half-plane, that is, with Re λi ≤ 0, i = 2, ... , n, is in particular realizable by a Toeplitz matrix.
Theorem 2.1. Let 0 1 0 ··· 0 . . . 0 0 .. .. . . . . . C = . . . . (1) . . . . 0 0 0 ··· 0 1 cn cn−1 ··· c2 c1 n n X n−i be the companion matrix of the polynomial p(x) = x − ci x . Then C is similar to a unit lower Hessenberg i=1 Toeplitz matrix T.
Proof. First, we shall assume that c1 = 0. De ne the n × n real matrix
1 0 ············ 0 . . 0 1 .. . . .. .. . s21 0 . . . ...... S = s s . . . . , (2) 31 22 . . . . . ...... s41 s32 . ...... . . . . 0 1 0 s(n−1)1 ··· s4(n−4) s3(n−3) s2(n−2) 0 1 where, for convenience, sij is the entry on the diagonal (j + i, j), i = 2, 3, ... , (n − 1); j = 1, 2, ... , (n − i). The entries of S−1 can be obtained in a straightforward manner. In fact, since det S = 1, then S−1 = adjS, and the entries (i, j) of S−1 are given by −1 i+j [S ]i,j = (−1) det bSji , (3) where bSji is the matrix S without the row j and the column i. We utilize MATLAB notation: for i < j, we let i : j denote the set {x ∈ Z : i ≤ x ≤ j} and A (i : j, k : l) denote the submatrix of A whose rows and columns are indexed by i : j and k : l, respectively. Then, det bSji = det S (j + 1 : i, j : i − 1) , i = 3, . . . n, j = 1, ... , i − 2. Thus, the formula (3) becomes h i S−1 = (−1)i+j det S (j + 1 : i, j : i − 1) , i = 3, . . . n, j = 1, ... , i − 2. (4) i,j
We consider the n × n matrix Toeplitz nonnegative realization of spectra via companion matrices Ë 233
0 1 0 ············ 0 | 0 | 0 s 0 1 0 ········· 0 | 0 | 0 21 −− −− −− −− −− −− −− −− | −− | −− | 0 | 0 . . | . | . . . | . | . S−1CS = , . . . . A | . | . . | 0 | . | 1 | 0 | 0 | 1 −− −− −− −− −− −− −− −− | −− | −− B | c2 − s2(n−2) | 0 where the entries on the block A are given by i−j+1 −1 X i−j−k+3 [S CS]ij = s(i−j+1)j + (−1) det S (j + k : i, j + k − 1 : i − 1) skj , k=0 i = 3, ... , n − 1, j = 1, ... , n − 2, with s0j = 1, s1j = 0, spj = 0 for p < 0, det S (1 : i, 0 : i − 1) = 0 and det S(h : g, h − 1 : g − 1) = 0 if h > g, while the entries on the block B are given by n−j+1 −1 X n−j+3−k [S CS]nj = ((−1) det S (j + k : n, j + k − 1 : n − 1) skj + ck s(n−j−k+1)j), k=0 −1 i = n, j = 1, ... , n − 2, with s0j = 1, s1j = 0, cl = 0 for l ≤ 1 and det S(i : j; k : l) = 0 if i > j. Then S CS has the form 0 1 0 ············ 0 . . . . ...... s21 . ...... s31 s22−s21 . . . . . . . . s −s s s −s s −s ...... 41 21 22 32 31 23 22 . . . . . (s51−s21s32 (s21s23−s41 . . . . . , s33−s32 . . . . . −s31s23) −s22s23+s42) ...... ...... 0 . . . . . ...... . . s3(n−3)−s3(n−4) s2(n−2)−s2(n−3) 1 ...... ······ c3−s3(n−3) c2−s2(n−2) 0 and it will be a Toeplitz matrix if all the entries in each diagonal under the main diagonal are equal. Then, −1 the entries sij of S are obtained by equating the entries in each diagonal under the main diagonal of S CS. Observe that on each one of these diagonals, each new entry from left to right contains exactly one new unknown. Then, the corresponding linear system always has a unique solution. Thus, our construction is unique. If tr(C) = c1 ≠ 0, then by taking 1 0 ········· 0 . . t1 1 0 ··· 0 .. . s11 1 . . . . t t .. .. . .. .. . 2 1 s21 s12 . . . . . . . S = , T = . . . . , . . . . . . . . 0 s s ...... 31 22 ...... . . . . . . . . . 1 ...... 0 tn ······ t2 t1 s(n−1)1 ··· s3(n−3) s2(n−2) s1(n−1) 1 234 Ë Macarena Collao, Mario Salas, and Ricardo L. Soto we also have similarity between C and T.
Remark 2.1. For our purpose, it is enough to prove the similarity between matrices C and T for the case tr(C) = 0 0. If C has spectrum Λ = {λ1, λ2, ... , λn} with c1 = tr(C) ≠ 0, we take the companion matrix C with spectrum 0 1 0 0 0 Λ = {λ1 − α, λ2 − α, ... , λn − α}, where α = n c1. Then tr(C ) = 0 and C is similar to a Toeplitz matrix T with tr(T0) = 0. Therefore T = T0 + αI is the required Toeplitz matrix, which is cospectral to C.
Example 2.1. Let us consider n = 5 with c1 = 0. Then we have
1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 −1 S = s21 0 1 0 0 , S = −s21 0 1 0 0 , (5) s31 s22 0 1 0 −s31 −s22 0 1 0 s41 s32 s23 0 1 s21s23 − s41 −s32 −s23 0 1 with 0 1 0 0 0 s 0 1 0 0 21 −1 s31 s22−s21 0 1 0 S CS = . s41−s21s22 s32−s31 s23−s22 0 1 (c5+c2s31+c3s21 (c4−s41+c2s22 c3−s32 c2−s23 0 −s21s32−s31s23) +s21s23−s22s23)
1 1 3 1 2 1 7 2 −1 If we take s21 = 4 c2, s22 = 2 c2, s23 = 4 c2, s31 = 3 c3, s32 = 3 c3, and s41 = 2 c4 + 32 c2, then S CS is the Toeplitz matrix 0 1 0 0 0 1 c 0 1 0 0 4 2 T= 1 1 , (6) 3 c3 4 c2 0 1 0 1 3 2 1 1 2 c4 + 32 c2 3 c3 4 c2 0 1 1 1 3 2 1 1 c5 + 6 c2c3 2 c4 + 32 c2 3 c3 4 c2 0 which is nonnegative if ci , i = 2, ... , 5, are nonnegative.
Theorem 2.2. If in Theorem 2.1 the companion matrix
0 1 0 ··· 0 . . . 0 0 .. .. . . . . . C = . . . . . . . . 0 0 0 ··· 0 1 cn cn−1 ··· c2 c1 is real nonnegative, then the corresponding Toeplitz matrix T is also real nonnegative.
Proof. In [11], the author proves the following formula, known as Trudi’s formula, for the n-th order determi- nant of a lower Hessenberg Toeplitz matrix t1 t0 0 ··· 0 . . . t t .. .. . 2 1 0 . . . . T = . . . . . (7) . . . . 0 ...... . . . . t0 tn ······ t2 t1 Toeplitz nonnegative realization of spectra via companion matrices Ë 235
That is, ! 0 X l1 + ··· + ln n−l1−···−ln l1 l2 ln det T = (−t0) t1 t2 ··· tn , (8) l1, ... , ln l1+2l2+···+nln=n where l1+···+ln = (l1+···+ln)! is the multinomial coe cient, and the sum is over the integer partitions of n. Let l1 ,...,ln l1!···ln! T = S−1CS be the matrix in Theorem 2.1. Since xI − T is also a lower Hessenberg Toeplitz matrix, we may compute, from (8), the characteristic polynomial of our Toeplitz matrix
0 1 0 ··· 0 . . . t 0 .. .. . 2 . . . . T = . . . . . (9) . . . . 0 ...... . . . . 1 tn ······ t2 0
That is, ! X l1 + ··· + ln l2 l3 ln l1 det(xI − T) = (−t2) (−t3) ··· (−tn) x . (10) l1, ... , ln l1+2l2+···+nln=n Now, we equate the coe cients of characteristic polynomials of T and C in (1),
n n−1 n−2 k det(xI − C) = x − c1x − c2x − · · · − cn−k x − · · · − cn−1x − cn . (11)
Then, ! X n − k + l2 + ··· + lk l2 l3 lk −ck = (−t2) (−t3) ··· (−tk) . (12) n − k, l2, ... , lk 2l2+···+klk=k
Observe that in each coe cient of the sum in (10), the term tk , with l1 = n − k, k = 2, 3, ... , n, appears isolated, that is, tk has exponent 1, and it is not a factor in a product with other factors ti , i ≠ k. Then, we may compute tk from (12), as 1 t = (c + g ), where (13) k n − k + 1 k k ! X n − k + l2 + ··· + lk−2 l2 lk−2 gk = (−t2) ··· (−tk−2) n − k, l2, ... , lk−2 2l2+···+(k−2)lk−2=k is a multivariate polynomial with positive coe cients. For example, for n = 5 we have: l1 + 2l2 + ··· + 5l5 = 5 with l1 l2 l3 l4 l5 5 0 0 0 0 3 1 0 0 0 2 0 1 0 0 1 2 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 0 1 Then
5 3 2 2 det(xI − T) = x − 4t2x − 3t3x + (3t2 − 2t4)x + 2t2t3 − t5
5 4 3 2 det(xI − C) = x − c1x − c2x − c3x − c4x − c5, and 2 c1 = 0; c2 = 4t2; c3 = 3t3; c4 = 2t4 − 3t2; c5 = t5 − 2t2t3 236 Ë Macarena Collao, Mario Salas, and Ricardo L. Soto
Thus, the rst terms tk are computed from (13) as 1 1 t = c , t = c , 2 n − 1 2 3 n − 2 3
1 n − 2 1 t = c + t2, t = c + (n − 3) t t . 4 n − 3 4 2 2 5 n − 4 5 2 3
1 (n − 3)(n − 4) n − 4 t = c − t3+ t2+(n − 4)t t 6 n − 5 6 6 2 2 3 2 4
l2 lk−2 Observe from (13), that tk ≥ 0 if the sum of exponents l2 + ··· + lk−2 in (−t2) ··· (−tk−2) is an even number, where one or more lj , j = 2, ... , k − 2, can be zero. If the sum of exponents is an odd number, we show that lj lm the sum is still nonnegative. In fact, observe that in tk , the negative terms of the form −tj tm with jlj + mlm = k lj lm and lj + lm being odd, are dominated by positive terms tj tm , obtained from terms of the form tj tk−j . In fact, since
j(lj − 1) + mlm = jlj + mlm − j = k − j,
lj−1 lm lj−1 lm lj lm it is clear that tk−j contains a positive term of the form tj tm , and then tj(tj tm ) = tj tm . Now, we show that the coe cient 1 n − k + j + lj − 1 + lm ! (n − k + 2) , (14) n − k + j + 1 (n − k + j)! lj − 1 !lm!
lj lm of the term tj tm obtained from tj tk−j is greater than or equal to the coe cient 1 n − k + l + lm ! j , (15) n − k + 1 (n − k)!lj!lm!
lj lm of the negative term −tj tm (lj + lm being odd). In fact, the term in (14) minus the term in (15) gives ! n − k + l + lm ! (n − k + 2) n − k + l + lm + 1 ··· (n − k + l + lm + j − 1) 1 j j j − (n − k + 1)!(lj − 1)!lm! (n − k + 2) (n − k + 3) ··· (n − k + j + 1) lj and ! n − k + l + lm ! n − k + l + lm + 1 ··· (n − k + l + lm + j − 1) 1 j j j − . (n − k + 1)!(lj − 1)!lm! (n − k + 3) ··· (n − k + j + 1) lj Observe that there are (j − 1) consecutive integer factors in the numerator and in the denominator of n − k + l + lm + 1 ··· (n − k + l + lm + j − 1) j j , (n − k + 3) ··· (n − k + j + 1) and since lj ≥ 1 and lj + lm ≥ 2, then n − k + l + lm + 1 ··· (n − k + l + lm + j − 1) 1 j j − ≥ 0. (n − k + 3) ··· (n − k + j + 1) lj l l l Thus, the coe cient in (14) dominates the coe cient in (15). For negative terms of the form −t j1 t j2 ··· t jm , j1 j2 jm lj lm with j1lj1 + ··· + jm ljm = k, we reduce these terms to terms of the form −tj tm , by replacing the last factor, and then we proceed as before. Thus, the sum in (13) does not contain negative addends and tk is nonnegative.
Example 2.2. From (13) we can express tk in terms of ck . For instance, 1 (n − 3)(n − 4) n − 4 t = c − t3+ t2+(n − 4)t t 6 n − 5 6 6 2 2 3 2 4
1 (n − 3)(n − 4) n − 4 1 n − 2 t = c − t3+ t2+(n − 4)t c + t2 6 n − 5 6 6 2 2 3 2 (n − 3) 4 2 2
1 (2n − 3)(n − 4) n − 4 n − 4 t = c + c3+ c2+ c c . 6 n − 5 6 6(n − 1)3 2 2(n − 2)2 3 (n − 1)(n − 3) 2 4 Toeplitz nonnegative realization of spectra via companion matrices Ë 237
As mentioned in the Introduction, it is well known that a list of complex numbers Λ = {λ1, ... , λn} is always −1 the spectrum of a Toeplitz matrix, indeed of a circulant matrix T = FDF , where D = diag{λ1, ... , λn} and F denotes the matrix of the discrete Fourier transform. Now, as a rst consequence of Theorem 2.1, we have the following simple result, in which the Toeplitz matrix T is not necessarily circulant. It is a unit lower Hessenberg Toeplitz matrix.
Corollary 2.1. Let Λ = {λ1, λ2, ... , λn} be a list of complex numbers. Then Λ is the spectrum of an n × n unit lower Hessenberg Toeplitz matrix, which is not necessarily circulant.
Proof. It is immediate from Theorem 2.1.
In [7, Theorem 3], Laey and Šmigoc solved the NIEP in the left-half plane. That is, they proved that if Λ = {λ1, ... , λn} is a list of complex numbers with λi ∈ H = {z ∈ C : Re z ≤ 0}, Λ = Λ, λ1 ≥ |λi| , i = 2, ... , n, then Λ is realizable if and only if n n X X 2 2 s1 = λi ≥ 0, s2 = λi ≥ 0, s1 ≤ ns2. i=1 i=1 Now we show that a realizable list of complex numbers in the left-half plane is in particular realizable by a Toeplitz matrix.
Corollary 2.2. Let Λ = {λ1, λ2, ... , λn}, with Re λi ≤ 0, i = 2, ... , n, be a realizable list of complex numbers. Then Λ is also realizable by a Toeplitz matrix.
Proof. If Λ is realizable, then from the result in [7], Λ is the spectrum of a matrix of the form C+ αI, where C is a nonnegative companion matrix with tr(C) = 0, α ≥ 0. Then, from Theorem 2.1 and Theorem 2.2, C is similar to a nonnegative Toeplitz matrix T with eigenvalues λ1 − α, λ2 − α, ... , λn − α. Hence T + αI is a nonnegative Toeplitz matrix with spectrum Λ.
Example 2.3. Let Λ = {13, −2 + 3i, −2 − 3i, −2 + 3i, −2 − 3i}. We construct a nonnegative Toeplitz matrix with spectrum Λ. Then we de ne P λ Γ = Λ − i = {12, −3 + 3i, −3 − 3i, −3 + 3i, −3 − 3i} n and compute the companion matrix of the polynomial P(x) = (x − 12)(x + 3 − 3i)2(x + 3 + 3i)2 = x5 − 72x3 − 648x2 − 2268x − 3888, that is, 0 1 0 0 0 0 0 1 0 0 C = 0 0 0 1 0 . 0 0 0 0 1 3888 2268 648 72 0 Now, from Theorem 2.1 and (6), we have the nonnegative Toeplitz matrix 0 1 0 0 0 18 0 1 0 0 0 T´ = 216 18 0 1 0 1620 216 18 0 1 11 664 1620 216 18 0 with spectrum Γ. Then T = T0 + I is the required nonnegative Toeplitz matrix. 238 Ë Macarena Collao, Mario Salas, and Ricardo L. Soto
Example 2.4. Consider Λ = {14, −2, −2, −1, −1, −1 ± 2i, −2 ± 3i, −1 ± 3i} . The characteristic polynomial as- sociated with Λ is
p(x) = x11 − 87x9 − 964x8 − 5779x7 − 23 492x6 − 68 857x5 −146 580x4 − 223 686x3 − 230 612x2 − 139 640x − 36 400, and then the corresponding companion matrix is nonnegative. By using the MATLAB routine presented in the
Appendix, Section 5, we compute the entries tk of our nonnegative Toeplitz matrix T with spectrum Λ : 87 964 53 149 162 164 t = , t = , t = , t = , 2 10 3 9 4 50 5 15
4463 925 637 82 990 237 2335 231 077 941 t = , t = , t = 6 40 500 7 75 8 216 000
27 636 959 655 994 3172 986 740 312 171 t = , t = , 9 273 375 10 3600 000
24 712 157 597 057 887 t = . 11 3645 000
In [9, Theorem 3] it was shown that the construction of the unit upper Hessenberg Toeplitz matrix is unique. Then, since our realizing Toeplitz matrices are unit lower Hessenberg Toeplitz, they are the same as the trans- pose of the unit upper Hessenberg Toeplitz matrices constructed in [9]. Therefore, our construction is also unique. However, our interest is the construction of nonnegative Toeplitz matrices with prescribed spectrum, which in our case become nonnegative unit lower Hessenberg Toeplitz ones.
Since the characteristic and minimal polynomials of companion matrices are equal, the following result is immediate:
Corollary 2.3. The Toeplitz matrices obtained from Theorem 2.1 have Jordan canonical form with only one Jordan block for each distinct eigenvalue.
3 Symmetric nonnegative block Toeplitz matrices with prescribed Jordan canonical form
We say that a spectrum is universally realizable (UR) if it is realizable for every possible Jordan canonical form (JCF) allowed by the spectrum. In this Section, we explore the universal realizability of lists which are realiz- able by symmetric nonnegative block Toeplitz matrices (they are not necessarily symmetric). Our results give su cient conditions for the existence of a symmetric nonnegative block Toeplitz matrix with prescribed JCF. First, we give some conditions for the existence of a symmetric nonnegative Toeplitz matrix with prescribed spectrum Λ = {λ1, ... , λn} for n = 2, 3. For n = 2 we have that Λ = {λ1, λ2} with λ1 ≥ |λ2| , λ1 > 0, is the spectrum of the symmetric nonnegative Toeplitz matrix " # 1 λ + λ λ − λ A = 1 2 1 2 . 2 λ1 − λ2 λ1 + λ2
For n = 3, we use a well known result due to Fiedler [4], which establishes that the real numbers λ1 ≥ λ2 ≥ λ3 and a1 ≥ a2 ≥ a3 ≥ 0 are, respectively, the eigenvalues and the diagonal entries of a 3 × 3 symmetric nonnegative matrix if and only if the vector of eigenvalues [λ1, λ2, λ3] majorizes the vector of diagonal entries [a1, a2, a3] and a1 ≥ λ2. Then, for our purpose, we have: Toeplitz nonnegative realization of spectra via companion matrices Ë 239
Lemma 3.1. The real numbers λ1 ≥ λ2 ≥ λ3, and a ≥ 0, are the eigenvalues and the diagonal entries, respec- tively, of a symmetric nonnegative Toeplitz matrix if and only if λ ≥ a 1 λ + λ ≥ 2a 1 2 (16) λ1 + λ2 + λ3 = 3a λ2 ≤ a.
In [17, Remark 3.3], the authors construct a 3 × 3 symmetric nonnegative matrix, which, in our case, after a row-column permutation, becomes the symmetric nonnegative Toeplitz matrix
a √s α − a 2 T = √s a √s , (17) 2 2 α − a √s a 2 p with spectrum Λ = {λ1, λ2, λ3}, where α = λ1 + λ2 − a and s = (λ1 − α)(λ1 − a), or the symmetric nonneg- ative Toeplitz matrix a √t β − a 2 T0 = √t a √t , (18) 2 2 β − a √t a 2 p with spectrum Λ = {λ1, λ2, λ3}, where β = λ1 + λ3 − a and t = (λ1 − β)(λ1 − a).
For real lists of the form Λ = {λ1, λ2, ... , λ2}, we have the following proposition:
Proposition 3.1. Let Λ = {λ1, λ2, ... , λ2} be a list of n real numbers with λ2 < 0 and λ1 + (n − 1)λ2 ≥ 0. Then Λ is the spectrum of a symmetric nonnegative Toeplitz matrix.
Proof. Without loss of generality we assume that λ1 + (n − 1)λ2 = 0. Consider the initial matrix λ1 0 ······ 0 . λ − λ λ 0 . 1 2 2 . . . . B = . . . . ∈ CSλ . . 0 . . . 1 ...... . . . . 0 λ1 − λ2 0 ··· 0 λ2
T h i T Let q = (n − 1)λ2, −λ2, ··· , −λ2 . Then from a well known result due to Brauer [1], A = B + eq is the required symmetric nonnegative Toeplitz matrix.
Let A be an n×n matrix with spectrum Λ = {λ1, ... , λn}, and let X be the n×r matrix, r < n, whose columns xi , kxik = 1, i = 1, ... , r, are eigenvectors of A corresponding to the eigenvalues λ1, ... , λr , with rank(X) = r. Let B = Ω+C be an r×r symmetric nonnegative matrix with spectrum Λ0 = {µ1, ... , µr}. Let Ω =diag{λ1, ... , λr} −1 h XT i and let S be an n × n orthogonal matrix partitioned as S = X | Y with S = Y T . Then, since AX = XΩ, we have " # " # ΩXT AY C 0 S−1AS = , S−1XCXT S = 0 Y T AY 0 0 " # BXT AY S−1 A + XCXT S = . (19) 0 Y T AY 240 Ë Macarena Collao, Mario Salas, and Ricardo L. Soto
Now, from a result in [12, Chapter VI, Lemma 1.2], if the matrices B and Y T AY in (19) have no common eigenvalues, then J(A + XCXT) = J(B) ⊕ J(Y T AY). Next, we apply a perturbation result given in [17], which introduces a symmetric version of a result due to Rado and published by Perfect in [13]. Rado’s result is a generalization of a Brauer’s result [1] and it shows how to change r eigenvalues of a matrix without changing any of its remaining (n − r) eigenvalues (see [16, 17] and [2, 18], respectively, about how Rado’s result has been applied to the NIEP and to the universal realizability problem). Thus we have:
Theorem 3.1. Let Λ = {λ1, λ2, ... , λn} be a list of complex numbers. Suppose that: i) There exists a partition Λ = Λ0 ∪ Λ1 ∪ ··· ∪ Λ1, where | {z } r times
Λ0 = {λ01, λ02, ... , λ0r} is a real list
Λ1 = {λ11, λ12, ... , λ1p}, Λ0 ∩ Λ1 = ∅, such that Γ1 = {λ, λ11, λ12, ... , λ1p}, 0 ≤ λ ≤ λ1, is the spectrum of a (p + 1) × (p + 1) nonnegative matrix with certain JCF. ii) There exists an r×r symmetric nonnegative Toeplitz matrix with spectrum Λ0 and diagonal entries λ, λ, ... , λ (r times). Then, there exists a symmetric nonnegative block Toeplitz matrix T with spectrum Λ and JCF preserving the JCF of Λ1.
Proof. From i) let A1 be a (p + 1) × (p + 1) nonnegative matrix with spectrum Γ1 and with certain prescribed JCF. Then, let A be the r(p + 1) × r(p + 1) block diagonal matrix A1 A 1 A = . .. . A1
Let A1x = λx with kxk = 1, where λ and x are the Perron eigenvalue and the Perron eigenvector of A1, respectively. From ii) let B be the r × r symmetric nonnegative Toeplitz matrix with spectrum Λ0 and diagonal entries λ, λ, ... , λ (r times). Let Ω be the r × r diagonal matrix Ω = diag{λ, λ, ... , λ}. Then for C = B − Ω, 0 c1 ··· cr−1 x 0 ··· 0 . . . . c 0 .. . 0 x .. . C = 1 , and X = , (20) . . . .. . .. .. . . 0 c1 . . . 0 cr−1 ··· c1 0 0 ... 0 x where X is the r(p + 1) × r matrix of normalized eigenvectors of A, it follows that
T T 0 c1xx ··· cr−1xx . . c xxT 0 .. . XCXT = 1 , (21) . . .. .. T . . . c1xx T T cr−1xx ··· c1xx 0 and from the symmetric Rado’s result,
T T A1 c1xx ··· cr−1xx . . c xxT A .. . T = A + XCXT = 1 1 . . .. .. T . . . c1xx T T cr−1xx ··· c1xx A1 Toeplitz nonnegative realization of spectra via companion matrices Ë 241 is a symmetric nonnegative block Toeplitz matrix with spectrum Λ. Since Λ0 ∩ Λ1 = ∅, T has a JCF, which preserves the JCF of Λ1.
Similar to Theorem 3.1, with obvious minor changes, the following result gives su cient conditions for the existence and construction of a symmetric nonnegative block Toeplitz matrix with prescribed spectrum.
Theorem 3.2. Let Λ = {λ1, λ2, ... , λn} be a list of complex numbers. Suppose that: i) There exists a partition Λ = Λ0 ∪ Λ1 ∪ ··· ∪ Λ1, where | {z } r times
Λ0 = {λ01, λ02, ... , λ0r} is a real list
Λ1 = {λ11, λ12, ... , λ1p}, such that Γ1 = {λ, λ11, λ12, ... , λ1p}, 0 ≤ λ ≤ λ1, is the spectrum of a (p + 1) × (p + 1) nonnegative matrix. ii) There exists an r×r symmetric nonnegative Toeplitz matrix with spectrum Λ0 and diagonal entries λ, λ, ... , λ (r times). Then, there exists a symmetric nonnegative block Toeplitz matrix T with spectrum Λ.
Example 3.1. Consider the list Λ = {5, 1, 0, −2, −2, −2} with the partition
Λ0 = {5, 1, 0}, Λ1 = {−2}, Γ1 = {2, −2}.
Then √1 0 0 0 2 0 0 0 0 2 √1 2 0 0 0 0 0 0 0 2 0 0 0 2 0 0 0 √1 0 2 A = , X = 1 . 0 0 2 0 0 0 0 √ 0 2 0 0 0 0 0 2 0 0 √1 2 0 0 0 0 2 0 0 0 √1 2 From (17) we compute a symmetric nonnegative Toeplitz matrix
q 3 2 2 2 q q B = 3 3 2 2 2 q 3 2 2 2
with eigenvalues 5, 1, 0 and diagonal entries 2, 2, 2. Then for C = B − diag{2, 2, 2}, √ √ 0 2 6 6 1 1 √4 √4 6 6 2 0 1 1 √ √ 4 4 √ √ 6 6 0 2 6 6 T = A + XCXT = √4 √4 √4 √4 6 6 6 6 2 0 4 4 √ √ 4 4 1 1 6 6 0 2 √4 √4 6 6 1 1 4 4 2 0 is a symmetric nonnegative block Toeplitz matrix with spectrum Λ.
It is easy to see that if T is obtained from Theorem 3.1, then we have the following necessary condition:
Proposition 3.2. Let T be an n × n symmetric nonnegative block Toeplitz matrix, obtained from Theorem 3.1, with spectrum Λ = {λ1, λ2, ... , λn} and r blocks of size t (n = rt). Then Λ has r lists of (t − 1) repeated eigenvalues. 242 Ë Macarena Collao, Mario Salas, and Ricardo L. Soto
Example 3.2. Consider the list
Λ = {9, 5, 4, −3 + 2i, −3 − 2i, −3 + 2i, −3 − 2i, −3 + 2i, −3 − 2i} with the partition
Λ0 = {9, 5, 4}, Λ1 = {−3 + 2i, −3 − 2i} , Γ1 = {6, −3 + 2i, −3 − 2i} .
Then q 3 0 1 0 6 2 2 q q T = 23 0 1 and B = 3 3 1 2 2 6 2 23 q 78 2 0 3 2 2 6 have, respectively, the spectra Γ and Λ . Moreover, T has a normalized Perron eigenvector √ 2 1, 6, 49 . 1 0 1 2549 2 T Let A = diag{T1, T1, T1}. Then the matrix T = A + XCX , √ √ √ 0 1 0 | 2 6 12 6 49 6 | 8 48 196 2549√ 2549√ 2549√ 2549 2549 2549 23 12 6 72 6 294 6 48 288 1176 0 1 | | 2 2549√ 2549√ 2549√ 2549 2549 2549 78 23 0 | 49 6 294 6 2401 6 | 196 1176 4802 2 2549 2549 5098 2549 2549 2549 −− −−−− −−−− − √ √ √ √ √ √ 2 6 12 6 49 6 | 0 1 0 | 2 6 12 6 49 6 2549√ 2549√ 2549√ 2549√ 2549√ 2549√ T = 12 6 72 6 294 6 23 12 6 72 6 294 6 , 2549 2549 2549 | 2 0 1 | 2549 2549 2549 √ √ √ √ √ √ 49 6 294 6 2401 6 | 78 23 0 | 49 6 294 6 2401 6 2549 2549 5098 2 2549 2549 5098 −− −−−− −−−− − √ √ √ 8 48 196 2 6 12 6 49 6 | | 0 1 0 2549 2549 2549 2549√ 2549√ 2549√ 48 288 1176 | 12 6 72 6 294 6 | 23 0 1 2549 2549 2549 2549√ 2549√ 2549√ 2 196 1176 4802 49 6 294 6 2401 6 23 2549 2549 2549 | 2549 2549 5098 | 78 2 0 is a symmetric nonnegative block Toeplitz with spectrum Λ.
Example 3.3. Let Λ = {20, 12, −3, −3, −3, −3, −5, −5, −5, −5}. i) First we compute a block symmetric nonnegative Toeplitz matrix with spectrum Λ and linear JCF. To do that we take the partition
Λ = Λ0 ∪ Λ1 ∪ Λ1 with
Λ0 = {20, 12}, Λ1 = {−3, −3, −5, −5}, Γ1 = {16, −3, −3 − 5, −5}.
Next, from the procedure in [15], we compute the nonnegative matrix
0 3 3 5 5 3 0 3 5 5 A1 = 3 3 0 5 5 5 3 3 0 5 5 3 3 5 0
1 with spectrum Γ1, Perron eigenvector x = kek e, and diagonal JCF. We also compute the symmetric nonnegative Toeplitz matrix " # 16 4 B = , 4 16 Toeplitz nonnegative realization of spectra via companion matrices Ë 243
with spectrum Λ0 and constant diagonal entry 16. Then
T T1 = A + XCX " # " #" #" T # A1 x 0 0 4 x 0 = + T A1 0 x 4 0 0 x 4 4 4 4 4 0 3 3 5 5 5 5 5 5 5 3 0 3 5 5 4 4 4 4 4 5 5 5 5 5 3 3 0 5 5 4 4 4 4 4 5 5 5 5 5 4 4 4 4 4 5 3 3 0 5 5 5 5 5 5 4 4 4 4 4 5 3 3 5 0 = 5 5 5 5 5 4 4 4 4 4 0 3 3 5 5 5 5 5 5 5 4 4 4 4 4 5 5 5 5 5 3 0 3 5 5 4 4 4 4 4 5 5 5 5 5 3 3 0 5 5 4 4 4 4 4 5 5 5 5 5 5 3 3 0 5 4 4 4 4 4 5 5 5 5 5 5 3 3 5 0 is a symmetric nonnegative block Toeplitz matrix with spectrum Λ and diagonal JCF. ii) With the same partition we compute, from the procedure in [18], the nonnegative matrix
0 3 3 5 5 4 0 2 5 5 A2 = 3 3 0 5 5 6 3 3 0 4 5 3 3 5 0 with JCFJ(A2) = diag{J1(16), J2(−3), J2(−5)}. Then " # A2 T T2 = + XCX A2 is a symmetric nonnegative block Toeplitz matrix with spectrum Λ and with JCFJ(T2) = diag{J1(20), J1(12), J2(−3), J2(−3), J2(−5), J2(−5)}. iii) In the same way, we may compute symmetric nonnegative block Toeplitz matrices T3 and T4 with Jor- dan blocks J2(−3), J2(−3), J1(−5), J1(−5), J1(−5), J1(−5) and J1(−3), J1(−3), J1(−3), J1(−3), J2(−5), J2(−5), respectively.
Appendix
In this Section, we present an algorithm that computes matrices C and T from Section 2. The informed ex- periment in Example 2.4 was performed using Matlab R2017a on a PC with a 3.6 GHz intel (R) Core (TM) i7 processor, with 8 GB of RAM and with operating system windows 10 Pro version 17134.829. The Partitions
Function nds the indices l2, l3, ... , lk−2 of the sum in (13) and it can be found in [3]. function [T,C]=comptoep(varargin)
% Algorithm to construct similar matrices C and T, where C is a companion % matrix and T is a Toeplitz matrix such that eig(C)=eig(T) % for a given set of eigenvalues defined by the user. % INPUT % - varargin is a variable-length input argument list that contains the % eigenvalues defined by the user as lambda_1, \lambda_2,...,\lambda_n. 244 Ë Macarena Collao, Mario Salas, and Ricardo L. Soto
% % OUTPUT % - C is a companion matrix whose eigenvalues are the list defined % by the user. % - T is a Toeplitz matrix which is similar to C.
% Author: Javier G. Pizarro % e-mail: [email protected] % Release date: 6/24/19 n=length(varargin); C=zeros(n,n); in=cell2mat(varargin); syms x % Constructing C cc=expand(prod(x-in)); cc=sym2poly(cc); C(n,:)=-cc(n+1:-1:2); C(1:n-1,2:n)=eye(n-1); % Start the proccess for constructing T cc=-cc(2:n+1); T=zeros(n,n); t=zeros(n,1); t(2)=cc(2)/(n-1); if n==2 T(2,1)=t(2); T(1,2)=1; return end t(3)=cc(3)/(n-2); if n==3 T(1,2)=1; T(2,3)=1; T(:,1)=t; T(3,2)=t(2); return end % For n>=4 start the computation of li’s %%% for i=4:n coef=partitions(i,2:i-2); [ncomb,nvar]=size(coef); for k=1:ncomb produ=1; for l=1:nvar produ=produ*(-t(l+1))^coef(k,l); end u=n-i; num=factorial(u+norm(coef(k,:),1)); den=factorial(u); for l=1:nvar den=den*factorial(coef(k,l)); end t(i)=t(i)+num*produ/den; end t(i)=(t(i)+cc(i))/(n-i+1); end Toeplitz nonnegative realization of spectra via companion matrices Ë 245 for ii=1:n T(ii:n,ii)=t(1:n+1-ii,1); end %Return T T(1:n-1,2:n)=T(1:n-1,2:n)+eye(n-1).
Acknowledgement: Supported by Fondecyt 1170313, Chile, Conicyt-PCHA/Doc.Nac./2016-21160684, Chile, and Universidad Católica del Norte.
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