Abstract
We consider the following inverse extreme eigenvalue problem: given the real numbers and the real vector , to construct a nonsymmetric tridiagonal matrix and a nonsymmetric arrow matrix such that are the minimal and the maximal eigenvalues of each one of their leading principal submatrices, and 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.
1. Introduction
We consider a particular inverse eigenvalue problem for real nonsymmetric tridiagonal matrices of the formand for real nonsymmetric arrow matrices of the formThis 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 , , of a matrix of form (1) or (2), together with an eigenvector of . 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 numbersand the vectorconstruct a matrix of form (1) or (2), such that and are, respectively, the minimal and the maximal eigenvalue of the leading principal submatrix , of , and is an eigenpair of .
It is known that a matrix of form (1) is diagonally similar to the symmetric irreducible tridiagonal matrixwhere , with , , and (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:where , with and ,
An important fact related to the above similarities is that they leave invariant the eigenvalues of the leading principal submatrices , . 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 an symmetric tridiagonal matrix of form (1) , such that and are, respectively, the minimal and the maximal eigenvalues of the leading principal submatrix of , , is
Lemma 3 (see [12]). Let be a matrix of form (2) with , . Let and , respectively, be the minimal and the maximal eigenvalue of the leading principal submatrix of , . Then and
Lemma 4 (see [7]). Let be a set of orthonormal eigenvectors associated with the eigenvalues of an matrix of form (2) with , , and with all its diagonal entries distinct, . Then for , where denotes the entry of the vector .
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]). Let be a monic polynomial of degree with all zeroes being real. If and are, respectively, the minimal and maximal zero of , then
(1) if , we have ,
(2) if , we have .
Lemma 6 (see [13]). Let be an nonsymmetric tridiagonal matrix of form (1), and let be the leading principal submatrix of , with characteristic polynomial , . Then the sequence satisfies the recurrence relation
Lemma 7. Let be an nonsymmetric arrow matrix of form (2), and let be the leading principal submatrix of , with characteristic polynomial , . Then, the sequence satisfies the recurrence relation
Proof. The result follows by expanding the determinants , .
2. Main Results
In this section we give a unique solution to Problem 1 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 and the vector be satisfyingandThen, there exists a unique nonsymmetric tridiagonal matrix of form (1), such that and are, respectively, the minimal and the maximal eigenvalue of the leading principal submatrix , , of , and is an eigenpair of .
Proof. Suppose satisfy (12), and the vector satisfies (13). To show the existence of a nonsymmetric tridiagonal matrix with the required properties is equivalent to show that the system of equationswhere , satisfies Lemma 6, has real solutions , and , , , with .
From Lemma 6 for , it follows that . Then, Now, from Lemma 6 for , system (14) can be written asFrom (19) and condition (13), it follows thatFrom (16) and (17) for we havefrom which andMoreover, from Lemma 5 and condition (12) we haveNow, from (20) and condition (13), we obtainFrom (16) and (17) it follows thatandFinally, from Lemma 5 and condition (12)for . Thus, we obtain a unique nonsymmetric tridiagonal matrix of form (1).
Theorem 9. Let the real numbers and the vector be satisfyingandThen there exists a unique nonsymmetric arrow matrix of form (2), such that and are, respectively, the minimal and the maximal eigenvalue of the leading principal submatrix , , of , and is an eigenpair of .
Proof. Suppose and satisfy conditions (31) and (32), respectively. To show the existence of a nonsymmetric arrow matrix with the required properties is equivalent to show that the system of equationswhere , satisfies Lemma 7, has real solutions , and , , , with .
From Lemma 7 for it follows that . Then From (41) and (42) for , it follows thatandNow, from conditions (32) and (44), we haveThus,Moreover, from Lemma 5 and condition (31),From Lemma 7, for , system (33) can be written asNow, from (41) and (42), we haveAnd from (44) and condition (32),Then,Finally, from Lemmas 3 and 5 and condition (31), we haveThus, we obtain a unique nonsymmetric arrow matrix of form (2).
We observe that the construction given by Theorems 8 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 numbersand the vectorsatisfy conditions (12) and (13). Our procedure from Theorem 8 gives the matrixwith the required spectral properties.
Example 2. The eigenvalues of matrixare given byAnd the components of an eigenvector associated with are given by the recurrence relation:In Table 1 we consider (52) with , , and . is the vector with components and the eigenvalues given in (53) and (54) and is the vector defined by (55). We denote by the constructed matrix by the procedure from Theorem 8 and is the vector with the extreme eigenvalues of . We consider the following expressions: , , and .
Example 3. Consider the -dimensional vector , whose entries are real numbers chosen randomly in nondecreasing order and a -dimensional vector , whose components are all nonzero. Let be nonsymmetric tridiagonal matrix constructed from the entries of in such a way that and are the minimal and maximal eigenvalues of the leading principal matrix of , . Let be the vector whose entries are the numbers and , , in nondecreasing order. Figure 1 shows the plot of and defined in Example 2, with and reconstructions of the matrix .
Example 4. In this example we consider random real numbers generated from the random Matlab function:andsatisfying conditions (31) and (32), respectively. From Theorem 9 procedure we obtain the matrixwith the required spectral properties.
Example 5. In Table 2, 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 the constructed matrix and is the vector with the extreme eigenvalues of . We consider the expressions, and .
Example 6. Consider dimensional vector , whose entries are real numbers chosen randomly in nondecreasing order and a -dimensional vector , whose components are all nonzero. Let be nonsymmetric arrow matrix constructed from the entries of in such a way that and are the minimal and maximal eigenvalues of the leading principal matrix of , . Let be the vector whose entries are numbers and , , in nondecreasing order. Figure 2 shows the plot of and defined in Example 5, with and reconstructions of the matrix .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was funded by Universidad de Tarapacá, Arica, Chile, Proyecto Mayor de Investigación Científica y Tecnológica UTA-4734-16 and Comisión Nacional de Investigación Científica y Tecnológica, Grant Fondecyt 1170313, Chile.