# Extreme Spectra Realization by Nonsymmetric Tridiagonal and Nonsymmetric Arrow Matrices

Authors: H. Pickmann-Soto; S. Arela-Pérez; Juan C. Egaña; Ricardo L. Soto
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

We consider the following inverse extreme eigenvalue problem: given the real numbers{λ1j,λjj}j=1n and the real vector x(n)=x1,x2,…,xn, to construct a nonsymmetric tridiagonal matrix and a nonsymmetric arrow matrix such that {λ1j,λjj}j=1n are the minimal and the maximal eigenvalues of each one of their leading principal submatrices, and x(n),λn(n) is an eigenpair of the matrix. We give sufficient conditions for the existence of such matrices. Moreover our results generate an algorithmic procedure to compute a unique solution matrix.


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## Body

## 1. Introduction

We consider a particular inverse eigenvalue problem for real nonsymmetric tridiagonal matrices of the form(1)A=a1b1c1a2b2c2a3⋱⋱⋱bn-1cn-1an;bici>0,i=1,2,…,n-1,and for real nonsymmetric arrow matrices of the form(2)B=a1b1b2⋯bn-1c1a2c2a3⋮⋱cn-1an;bici>0,i=1,2,…,n-1.This kind of matrices appears in several areas of science and engineering, as in the Lanczos method for tridiagonalizing a nonsymmetric matrix or for computing the Gaussian quadrature [1–3]. The nonsymmetric arrow matrices used to be an important tool for computing eigenvalues via dividing and conquering approximations, in the study of nonsymmetric eigenvalue problem [4]. The symmetric inverse eigenvalue problem has attracted the attention of many authors. In contrast, the nonsymmetric case has been less studied [5, 6]. In this paper we discuss the inverse eigenvalues problem for matrices (1) and (2) considering the following spectral information: the set of minimal and maximal eigenvalues of all leading principal submatrices Aj, j=1,2,…,n, of a matrix A of form (1) or (2), together with an eigenvector of A. This type of spectral information has been recently considered in the literature [7–10]. More precisely, we consider the following problem.Problem 1.
Given the list of real numbers(3)λ1n,λ1n-1,…,λ12,λ11,λ22,…,λn-1n-1,λnn,and the vector(4)xn=x1,x2,…,xn,construct a matrix A of form (1) or (2), such that λ1j and λjj are, respectively, the minimal and the maximal eigenvalue of the leading principal submatrix Aj, j=1,2,…,n of A, and x(n),λnn is an eigenpair of A.It is known that a matrix of form (1) is diagonally similar to the symmetric irreducible tridiagonal matrix(5)DAD-1=a1b1c1b1c1a2b2c2b2c2⋱⋱⋱an-1bn-1cn-1bn-1cn-1an,where D=diag⁡γ1,γ2,…,γn, with γi=(cici+1⋯cn-1)/(bibi+1⋯bn-1), i=1,…,n-1, and γn=1 (see [2]).In the same way, it can be determined that a matrix of form (2) is diagonally similar to the symmetric irreducible arrow matrix:(6)DBD-1=a1b1c1b2c2⋯bn-1cn-1b1c1a2b2c2a3⋮⋱bn-1cn-1an,where D=diag⁡δ1,δ2,…,δn, with δ1=1 and δi+1=bi/ci, i=1,…,n-1.An important fact related to the above similarities is that they leave invariant the eigenvalues of the leading principal submatricesAj, j=1,2,…,n. Then the following results in [7, 11, 12] hold for matrices of forms (1) and (2) as well.Lemma 2 (see [11]).
A necessary and sufficient condition for the existence of ann×n symmetric tridiagonal matrix of form (1) (bi=ci), such that λ1(j) and λj(j) are, respectively, the minimal and the maximal eigenvalues of the leading principal submatrix Aj of A, j=1,2,…,n, is (7)λ1n<λ1n-1<⋯<λ12<λ11<λ22<⋯<λn-1n-1<λnn.Lemma 3 (see [12]).
LetA be a matrix of form (2) with bi=ci≠0, i=1,…,n-1. Let λ1(j) and λj(j), respectively, be the minimal and the maximal eigenvalue of the leading principal submatrix Aj of A, j=1,2,…,n. Then(8)λ1j<⋯<λ13<λ12<λ11<λ22<λ33<⋯<λjj, and (9)λ1j<ai<λjj,i=2,3,…,j,foreachj=2,3,…,n.Lemma 4 (see [7]).
Letx1,x2,…,xn be a set of orthonormal eigenvectors associated with the eigenvalues λ1,λ2,…,λn of an n×n matrix A of form (2) with bi=ci>0, i=1,…,n-1, and with all its diagonal entries aj distinct, j=2,3,…,n. Then xμj≠0 for μ,j=1,2,…,n, where xμj denotes the μth entry of the vector xj.In [11, 12], the authors show how to construct symmetric tridiagonal and symmetric arrow matrices from the spectral information (3). Then, if the given spectral information is only (3), we may construct matrices of form (1) and (2), respectively, from the symmetric matrices (5) and (6), by similarity. However, these constructions are not unique. In order to obtain a unique solution, we consider the spectral information (3) and (4). We shall need the following known results.Lemma 5 (see [12]).
LetP(λ) be a monic polynomial of degree n with all zeroes being real. If λ1 and λn are, respectively, the minimal and maximal zero of Pλ, then
(1) ifμ<λ1, we have -1nPμ>0,
(2) ifμ>λn, we have Pμ>0.Lemma 6 (see [13]).
LetA be an n×n nonsymmetric tridiagonal matrix of form (1), and let Aj be the j×j leading principal submatrix of A, with characteristic polynomial Pjλ=detλIj-Aj, j=1,2,…,n. Then the sequence Pjλj=1n satisfies the recurrence relation(10)P0λ=1,P1λ=λ-a1,Pjλ=λ-ajPj-1λ-bj-1cj-1Pj-2λ,j=2,3,…,n,Lemma 7.
LetA be an n×n nonsymmetric arrow matrix of form (2), and let Aj be the j×j leading principal submatrix of A, with characteristic polynomial Pjλ=detλIj-Aj, j=1,2,…,n. Then, the sequence Pjλj=1n satisfies the recurrence relation (11)P1λ=λ-a1P2λ=λ-a2P1λ-b1c1Pjλ=λ-ajPj-1λ-bj-1cj-1∏i=2j-1λ-ai,j=3,4,…,n.Proof.
The result follows by expanding the determinantsdet(λIj-Aj), j=1,2,…,n.


## 2. Main Results

In this section we give a unique solution to Problem1 for the matrices of forms(1) and (2). Conditions (12) and (13), as well as conditions (31) and (32) below, arise from Lemmas 2, 3, and 4 by the similarity of matrices (1) and (5), as well as matrices (2) and (6).Theorem 8.
Let the real numbers{λ1j,λjj}j=1n and the vector x(n)=x1,x2,…,xn be satisfying(12)λ1n<λ1n-1<⋯<λ12<λ11<λ22<⋯<λn-1n-1<λnnand(13)xi≠0,i=1,2,…,n.Then, there exists a unique nonsymmetric tridiagonal matrix A of form (1), such that λ1j and λjj are, respectively, the minimal and the maximal eigenvalue of the leading principal submatrix Aj, j=1,2,…,n, of A, and x(n),λnn is an eigenpair of A.Proof.
Suppose{λ1j,λjj}j=1n satisfy (12), and the vector x(n) satisfies (13). To show the existence of a nonsymmetric tridiagonal matrix A with the required properties is equivalent to show that the system of equations(14)Pjλij=0;j=1,2,…,n,i=1,jAxn=λnnxn,where Pjλ=detλIj-Aj, j=1,2,…,n satisfies Lemma 6, has real solutions aj, j=1,2,…,n and bi, ci, i=1,2,…,n-1, with cibi>0.
From Lemma6 for j=1, it follows that P1λ11=λ11-a1=0. Then, (15)a1=λ11. Now, from Lemma 6 for j=2,3,…,n, system (14) can be written as(16)Pjλ1j=λ1j-ajPj-1λ1j-bj-1cj-1Pj-2λ1j=0,(17)Pjλjj=λjj-ajPj-1λjj-bj-1cj-1Pj-2λjj=0,(18)j=2,3,…,n.(19)a1x1+b1x2=λnnx1(20)cj-1xj-1+ajxj+bjxj+1=λnnxj,j=2,…,n-1(21)cn-1xn-1+anxn=λnnxn.
From (19) and condition (13), it follows that(22)b1=λnn-a1x1x2.From (16) and (17) for j=2 we have(23)P2λ12=λ12-a2P1λ12-b1c1P0λ12=0P2λ22=λ22-a2P1λ22-b1c1P0λ22=0,from which (24)c1=1b1λ22-λ12P1λ12P1λ22P1λ12-P1λ22.and(25)a2=λ12P1λ12-λ22P1λ22P1λ12-P1λ22.Moreover, from Lemma 5 and condition (12) we have(26)b1c1=-1λ22-λ12P1λ12P1λ22-1P1λ12-P1λ22>0.Now, from (20) and condition (13), we obtain(27)bj=λnn-ajxjxj+1-cj-1xj-1xj+1;j=2,3,…,n-1.From (16) and (17) it follows that(28)cj-1=1bj-1λjj-λ1jPj-1λ1jPj-1λjjPj-1λ1jPj-2λjj-Pj-1λjjPj-2λ1j,j=3,…,n.and(29)aj=λ1jPj-1λ1jPj-2λjj-λjjPj-1λjjPj-2λ1jPj-1λ1jPj-2λjj-Pj-1λjjPj-2λ1j,j=3,…,n.Finally, from Lemma 5 and condition (12)(30)bj-1cj-1=-1j-1λjj-λ1jPj-1λ1jPj-1λjj-1j-1Pj-1λ1jPj-2λjj-Pj-1λjjPj-2λ1j>0,for j=3,4,…,n. Thus, we obtain a unique nonsymmetric tridiagonal matrix of form(1).Theorem 9.
Let the real numbers{λ1j,λjj}j=1n and the vector x(n)=x1,x2,…,xn be satisfying(31)λ1n<λ1n-1<⋯<λ12<λ11<λ22<⋯<λn-1n-1<λnnand(32)xi≠0,i=1,2,…,n.Then there exists a unique nonsymmetric arrow matrix A of form (2), such that λ1j and λjj are, respectively, the minimal and the maximal eigenvalue of the leading principal submatrix Aj, j=1,2,…,n, of A, and x(n),λnn is an eigenpair of A.Proof.
Suppose{λ1j,λjj}j=1n and x(n) satisfy conditions (31) and (32), respectively. To show the existence of a nonsymmetric arrow matrix A with the required properties is equivalent to show that the system of equations(33)Pjλij=0,j=1,2,…,n,i=1,j(34)Axn=λnnxn,where Pjλ=detλIj-Aj, j=1,2,…,n satisfies Lemma 7, has real solutions aj, j=1,2,…,n and bi, ci, i=1,2,…,n-1, with bici>0.
From Lemma7 for j=1 it follows that P1λ11=λ11-a1=0. Then (35)a1=λ11From (41) and (42) for j=2, it follows that(36)P2λ12=λ12-a2P1λ12-b1c1=0P2λ22=λ22-a2P1λ22-b1c1=0and(37)a2=λ12P1λ12-λ22P1λ22P1λ12-P1λ22.Now, from conditions (32) and (44), we have(38)c1=λnn-a2x2x1.Thus,(39)b1=1c1λ22-λ12P1λ12P1λ22P1λ12-P1λ22.Moreover, from Lemma 5 and condition (31),(40)b1c1=-1λ22-λ12P1λ12P1λ22-1P1λ12-P1λ22>0
From Lemma7, for j=3,4,…,n, system (33) can be written as(41)Pjλ1j=λ1j-ajPj-1λ1j-bj-1cj-1∏i=2j-1λ1j-ai=0,(42)Pjλjj=λjj-ajPj-1λjj-bj-1cj-1∏i=2j-1λjj-ai=0,(43)a1x1+∑k=1n-1bkxk+1=λnnx1(44)cj-1x1+ajxj=λnnxjj=2,3,…,n.Now, from (41) and (42), we have(45)aj=λ1jPj-1λ1j∏i=2j-1λjj-ai-λjjPj-1λjj∏i=2j-1λ1j-aiPj-1λ1j∏i=2j-1λjj-ai-Pj-1λjj∏i=2j-1λ1j-ai.And from (44) and condition (32),(46)cj-1=λnn-ajxjx1.Then,(47)bj-1=1cj-1λjj-λ1jPj-1λ1jPj-1λjjPj-1λ1j∏i=2j-1λjj-ai-Pj-1λjj∏i=2j-1λ1j-ai.Finally, from Lemmas 3 and 5 and condition (31), we have(48)bj-1cj-1=-1j-1λjj-λ1jPj-1λ1jPj-1λjj-1j-1Pj-1λ1j∏i=2j-1λjj-ai-Pj-1λjj∏i=2j-1λ1j-ai>0.Thus, we obtain a unique nonsymmetric arrow matrix of form(2).We observe that the construction given by Theorems8 and 9 generalizes the procedures given in [11, 12], in which the authors only consider the extremal eigenvalues as initial spectral information.


## 3. Numerical Examples

Example 1.
The real numbers(49)λ16λ15λ14λ13λ12λ11λ22λ33λ44λ55λ66-8-5-3-2-126791215,and the vector(50)x=0.1,0.2,0.3,0.4,0.5,0.6satisfy conditions (12) and (13). Our procedure from Theorem 8 gives the matrix(51)A=2.00006.500000001.84623.00007.384600001.08332.00009.208300002.34894.78656.761500006.71222.55155.8989000012.88764.2603,with the required spectral properties.Example 2.
The eigenvalues of matrix(52)A=αβ0⋯0γαβ⋯⋮0γα⋱0⋮⋮⋱⋱β0⋯0γα;βγ>0,are given by(53)λ1j=α+2βγcos⁡πj+1,j=2,3,…,n(54)λjj=α+2βγcos⁡jπj+1,j=1,2,…,n.And the components of an eigenvector associated with λnn are given by the recurrence relation:(55)x1=1;x2=1βλnn-α;xi=1βλnn-αxi-1-γxi-2,i=3,…,n
In Table1 we consider A (52) with α=4, β=1, and γ=2. λ is the vector with components and the eigenvalues given in (53) and (54) and x is the vector defined by (55). We denote by A~ the constructed matrix by the procedure from Theorem 8 and λ~ is the vector with the extreme eigenvalues of A~. We consider the following expressions: eA=log‖A-A~‖/‖A‖, eλ=log⁡‖λ-λ~‖/‖λ‖, and ex=log⁡A~x-λ~nnx/λnnx.Table 1 n e A e λ e x 5 - 14.8638 - 15.7126 - 15.5670 10 - 13.8739 - 14.9518 - 15.0035 15 - 13.1475 - 14.6297 - 14.3310 20 - 12.7062 - 14.7679 - 14.8618 25 - 12.1855 - 14.3766 - 14.1447 30 - 12.2131 - 13.8168 - 13.4682 40 - 12.0392 - 12.9129 - 12.2304 50 - 11.4974 - 11.8000 - 11.2788Example 3.
Consider the(2n-1)-dimensional vector λ, whose entries are real numbers chosen randomly in nondecreasing order and a n-dimensional vector x, whose components are all nonzero. Let A~ be n×n nonsymmetric tridiagonal matrix constructed from the entries of λ in such a way that λ~1(j) and λ~j(j) are the minimal and maximal eigenvalues of the j×j leading principal matrix A~j of A~, j=1,2,…,n. Let λ~ be the vector whose entries are the numbers λ~1(j) and λ~j(j), j=1,2,…,n, in nondecreasing order. Figure 1 shows the plot of eλ and ex defined in Example 2, with n=50 and 100 reconstructions of the matrix A~.Figure 1Example 4.
In this example we consider random real numbers generated from the random Matlab function:(56)λ17λ16λ15λ14λ13λ12-5.6310-5.2374-4.6520-4.3671-4.3250-3.7990λ11λ22λ33λ44λ55λ66λ77-1.60360.14004.57344.57705.90976.47806.6793and(57)x=0.5886,0.3527,-0.1856,0.0370,-0.4738,0.3565,-0.3765satisfying conditions (31) and (32), respectively. From Theorem 9 procedure we obtain the matrix(58)A=-1.60360.7312-7.05340.2249-3.11323.8302-1.21975.2347-2.0554-1.10193.18550.5884-2.6848-1.55574.74662.12993.1628-3.53681.1509with the required spectral properties.Example 5.
In Table2, we construct nonsymmetric arrow matrices of different orders, from data obtained arbitrarily by the random function in Matlab, which satisfy conditions (31) and (32) of Theorem 9. We denote by A~ the constructed matrix and λ~ is the vector with the extreme eigenvalues of A~. We consider the expressions, eλ=log⁡λ-λ~/λ and ex=log⁡A~x-λ~n(n)x/λn(n)x.Table 2 n e λ e x 5 - 14.9841 - 15.0703 10 - 14.9891 - 15.9881 15 - 14.9286 - 15.7874 20 - 14.8021 - 15.9188 25 - 14.7349 - 15.4561 30 - 14.8234 - 14.5340 40 - 14.6183 - 14.2118 50 - 14.6217 - 14.3512Example 6.
Consider(2n-1)-dimensional vector λ, whose entries are real numbers chosen randomly in nondecreasing order and a n-dimensional vector x, whose components are all nonzero. Let A~ be n×n nonsymmetric arrow matrix constructed from the entries of λ in such a way that λ~1(j) and λ~j(j) are the minimal and maximal eigenvalues of the j×j leading principal matrix A~j of A~, j=1,2,…,n. Let λ~ be the vector whose entries are numbers λ~1(j) and λ~j(j), j=1,2,…,n, in nondecreasing order. Figure 2 shows the plot of eλ and ex defined in Example 5, with n=50 and 100 reconstructions of the matrix A~.Figure 2



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