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ABSTRACT
In this paper, we investigate a generalized inverse eigenproblem for pseudo-Jacobi matrices with mixed eigendata. These matrices appear in non-Hermitian Quantum Mechanics and extend the well-known concept of Jacobi matrices. It is shown that a unique pseudo-Jacobi matrix may be recovered from certain prescribed mixed eigendata, i.e. its leading principal submatrix, two distinct real eigenvalues, and part of the corresponding eigenvectors. An algorithm is provided for the reconstruction of such a matrix, and illustrative numerical experiments are presented to test the algorithm. The recurrence relation involving leading principal minors is crucial for the problem solution.
1. Introduction
In this paper, we shall be concerned with the set of real tridiagonal matrices
with
,
positive and satisfying
,
,
and 0<r<n. This set is denoted by
, and its elements are known as pseudo-Jacobi matrices (see the definition in [Citation1]). If the values of
and
,
in the above matrix T are equal and positive respectively, then T is changed into a Jacobi matrix. These matrices attracted the attention of scientists as they appear in different areas of science, such as mechanics and theoretical physics (e.g. see [Citation1–4] and the references therein). In particular, the literature on inverse eigenvalue problems for Jacobi matrices is very rich. Much of the basic theory on existence and uniqueness of solution of these problems is due to Hochstadt [Citation5] and Hald [Citation6], and numerical algorithms for reconstructing Jacobi matrices from prescribed spectral data were proposed by Boley and Golub [Citation7], Ferguson [Citation8], etc. For other results, we refer the readers to [Citation9–15]. Inverse eigenproblems have many applications in vibrating systems [Citation12], classical moment problems [Citation2], etc. The study of inverse eigenproblem for pseudo-Jacobi matrices also deserves attention since these matrices arise in connection with the discretization and truncation of the Schrödinger equation in non-Hermitian quantum mechanics [Citation1] (see also [Citation3,Citation16,Citation17]). In addition, generalized inverse eigenproblems on the reconstruction of a Jacobi matrix and a symmetric tridiagonal matrix from its leading principal submatrix, two distinct real eigenvalues, and part of the corresponding eigenvectors, were presented, respectively in [Citation18,Citation19]. In this article, we will consider the generalized inverse eigenproblem with mixed eigendata in the pseudo-Jacobi case.
Consider the eigenvalue problem
with
and
being the non-Hermitian matrix
(1)
(1) where the pair
is said to be an eigenpair of
The goal of this work is to solve the generalized inverse eigenproblem stated below, denoted by the acronym of generalized inverse eigenproblem for pseudo-Jacobi matrices (GIEPPJ), consisting in reconstructing a matrix in
from the knowledge of partial spectral data.
The GIEPPJ statement is as follows.
GIEPPJ: Given as in (Equation1
(1)
(1) ), an
Jacobi matrix
, real distinct numbers λ and μ, real vectors
,
, where 0<r<n, find nonzero real vectors
,
and a pseudo-Jacobi matrix
such that
is its
leading principal submatrix, and
and
are eigenpairs of
, i.e.
and
, where
and
.
The rest of this paper is organized as follows. In Section 2, the eigenvectors X and Y of the matrix pair are characterized. In Section 3, a necessary and sufficient condition for the existence of a unique solution for the GIEPPJ is presented. In Section 4, an algorithm is proposed to reconstruct the pseudo-Jacobi matrix in
, and examples are given to illustrate the algorithm. Finally, in Section 5 some conclusions are presented.
2. Eigenvectors of the matrix pair ![](//:0)
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Throughout this paper, let denote the set of eigenvalues of the matrix pair
, and the determination of these pairs is the direct generalized spectral problem. Let
be the ith leading principal minor of
, where
and
are, respectively, the
leading principal submatrix of matrices
and
for
. Assume that
and
. Let us define:
(2)
(2)
(3)
(3)
(4)
(4)
(5)
(5) where λ, μ,
and
are as in the GIEPPJ.
The following three-term recurrence relation holds:
(6)
(6)
Before stating our main result we present preliminary lemmas.
Lemma 2.1
Let be an eigenpair of the matrix pair
where
. If
for
then
two consecutive components of X cannot simultaneously vanish;
,
.
Proof.
(a), (b) Assume that is an eigenpair of
,
and write this matrix equation as the system:
(7)
(7) where
and
are interpreted as zero.
If for
, then by the first equation in (Equation7
(7)
(7) ) we conclude that
implies
. By the second equation, taking
implies
, and successively, considering the jth equation (Equation7
(7)
(7) ), we find that
implies
. So, up to the
th equation in (Equation7
(7)
(7) ),
implies that
. Conversely,
implies
, and all the components of X must vanish, contradicting that
. Then (a) and (b) hold.
(c) Writing
and multiplying the first, second and third equations in (Equation7
(7)
(7) ), respectively, by
we obtain
(8)
(8) where
.
Comparing (Equation8(8)
(8) ) with (Equation6
(6)
(6) ), we conclude that
,
, where K is a nonzero constant. If
, then
. Thus, we get
Therefore, if for any
,
for any
.
An immediate consequence of Lemma 2.1 is the following.
Lemma 2.2
Let be an eigenpair of
. Assume that
is a nonzero vector. If
for
then
and
Lemma 2.3
Under the assumptions in the GIEPPJ and Lemma 2.2, and assuming that we have:
If
then
and
If
then
and
where
is an arbitrary nonzero real number.
Proof.
(1) Since , then
. By Lemma 2.2 we have
. If
for
, from the proof of Lemma 2.1 we conclude that
. Therefore,
and
From the first equation in (Equation7
(7)
(7) ), as
, we obtain by using the value of
The proof proceeds by induction. We assume
and we show that
Using the recurrence relation
in (Equation7
(7)
(7) ), as
, we obtain
Hence (1) holds.
(2) If , then
. If
for
, from the proof of Lemma 2.1 we get
, and we easily obtain
Since , by Lemma 2.2, we may consider
as an arbitrary nonzero real number, and the result follows.
Lemma 2.4
Under the hypothesis of Lemma 2.3, we have:
Proof.
For r=1, the statement is trivial. For r=2, we get
The proof follows by induction. We assume the validity of the statement for r=p, , i.e.
By considering the partition
where
, we obtain
This shows the validity of the statement for r = p+1, and so the statement is true for
.
Lemma 2.5
Under the hypothesis of Lemma 2.3, the following holds:
If
then
If
and
then
If
and
then
If
then
where and
are arbitrary nonzero real numbers.
Proof.
(1) If , then
and
. By Lemmas 2.3 and 2.4, we obtain
(2) If and
, then
and
. Thus
. By Lemmas 2.3 and 2.4, we have
where
is an arbitrary nonzero real number.
(3) If and
, then
,
and
. Thus, by Lemmas 2.3 and 2.4 we get
where
is an arbitrary nonzero real number.
(4) If , then
and
. Therefore, by Lemmas 2.3 and 2.4 we find,
and the proof is complete.
3. Solution of the GIEPPJ
Assume that and
are eigenpairs of
. Thus,
(9)
(9) Let
,
be the
trailing principal submatrix of
,
in lines
, and consider the partition
where
and
. The system of matrix equations (Equation9
(9)
(9) ) can be equivalently expressed as follows:
(10)
(10)
(11)
(11)
(12)
(12)
(13)
(13)
(14)
(14)
(15)
(15)
We intend to find a necessary and sufficient condition for the existence of a unique solution of the GIEPPJ. The solution of the GIEPPJ is unique if and only if there exists a unique positive number , unique nonzero real vectors
,
with
, and a unique Jacobi matrix
satisfying Equations (Equation10
(10)
(10) )–(Equation15
(15)
(15) ).
Theorem 3.1
There exists a unique triplet with
solving the GIEPPJ if and only if the following conditions are satisfied:
and
for any
One of the following holds:
where
for any
where the parameters
and
are defined in (Equation3
(3)
(3) )–(Equation5
(5)
(5) ).
Proof.
Firstly, we prove the sufficiency. Under the condition (1), the results in Lemmas 2.2–2.5 hold. Under the condition (2), then and
, and this imply that Equations (Equation10
(10)
(10) ) and (Equation12
(12)
(12) ) have unique solutions
and
. From the first part of Lemma 2.3, we obtain the unique expressions of
and
. Meanwhile,
and
.
As
and
subtracting (Equation15
(15)
(15) ) from (Equation14
(14)
(14) ) yields
Since the condition (3) holds, then . By substituting in the above equation the unique values of
and
provided by the first part of Lemma 2.3, and the unique value of
in the first part of Lemma 2.5, we obtain the following quadratic equation
(16)
(16) in the unknown
, where
and
are given in the condition (3). Let Δ be the discriminant of Equation (Equation16
(16)
(16) ).
If ,
,
, then Equation (Equation16
(16)
(16) ) has a double positive root. In this case,
.
If ,
, then Equation (Equation16
(16)
(16) ) has a positive and a negative root, and we consider the positive root
.
If ,
,
, then Equation (Equation16
(16)
(16) ) has a positive and a zero root, and we take the positive root
.
Therefore, the condition (3) ensures that Equation (Equation16(16)
(16) ) has a unique positive root. Since
and
, the solutions of Equations (Equation10
(10)
(10) ) and (Equation12
(12)
(12) ) are both nonzero, that is
and
in the first part of Lemma 2.3.
Now, by Equation (Equation11(11)
(11) ) we obtain
(17)
(17)
(18)
(18)
(19)
(19) Similarly, by Equation (Equation13
(13)
(13) ) we find
(20)
(20)
(21)
(21)
(22)
(22) Eliminating
from Equations (Equation18
(18)
(18) ) and (Equation21
(21)
(21) ) yields
(23)
(23) for
. Next, by eliminating
from Equations (Equation19
(19)
(19) ) and (Equation22
(22)
(22) ), we find
(24)
(24) Hence summing Equation (Equation23
(23)
(23) ) over
, we get
(25)
(25) Further, summing Equations (Equation24
(24)
(24) ) and (Equation25
(25)
(25) ) yields
(26)
(26) Equation (Equation26
(26)
(26) ) and the condition (4) imply that
exists uniquely and can be expressed as
(27)
(27) Since
and
, from Equations (Equation17
(17)
(17) ) and (Equation20
(20)
(20) ) we have
(28)
(28) The above values of
are equal, and
are uniquely determined. As
, then
and
cannot be simultaneously zero. This implies that
is unique for
. Thus, from Equations (Equation18
(18)
(18) ) and (Equation21
(21)
(21) ) we get
(29)
(29) From Equations (Equation19
(19)
(19) ) and (Equation22
(22)
(22) ),
can be expressed as
(30)
(30) In addition, if
, then the above two values of
are equal, and all the
are unique. Thus, the positive number
and the truncated matrix
are unique.
Now we prove the necessity. Assume the GIEPPJ has a unique solution for the triplet , where
, i.e. Equations (Equation10
(10)
(10) )–(Equation15
(15)
(15) ) have unique solutions. Because Equations (Equation10
(10)
(10) ) and (Equation12
(12)
(12) ) have unique solutions, then
and
, i.e.
. Since
and
, then
and
, otherwise, from Equations (Equation10
(10)
(10) ) and (Equation12
(12)
(12) ) we would obtain
, a contradiction. Moreover,
implies that
, because
from the first part of Lemma 2.3. Hence there exist
and
for
. Since
exists uniquely and Equation (Equation16
(16)
(16) ) can be obtained from Equations (Equation14
(14)
(14) ) and (Equation15
(15)
(15) ), and the first part of Lemmas 2.3 and 2.5, then the condition (3) holds. Finally, the condition (4) is satisfied because Equation (Equation26
(26)
(26) ) holds for all
and
exists uniquely for any
.
4. Numerical algorithm and experiments
Based on Theorem 3.1 we present the following algorithm to solve the GIEPPJ.
By using MATLAB R2013a with a machine precision of around , we give the following example to illustrate Algorithm 1.
Example 4.1
Let ,
,
By using Algorithm 1, the four conditions in Theorem 3.1 hold. Then we get
and
Moreover, let and
. We can reconfirm that
,
,
and
. That is to say
and
are the eigenpairs of
.
In the following, we present an example to test Algorithm 1 in a different way.
Example 4.2
Given the matrix as in (Equation1
(1)
(1) ) with entries
Let be the rth leading principal matrix of
with entries
Assume λ and μ are the first two real and distinct generalized eigenvalues of the matrix pair ,
and
are their corresponding generalized eigenvectors. Take the partition
where
. By using Algorithm 1, we obtain the pseudo-Jacobi matrix
and real eigenvectors
and
. Let
and
. In Table , we show that
and
are eigenpairs of
. On the other hand,
and
are the unique nonzero real vectors that can be obtained as well as
is the unique recovered pseudo-Jacobi matrix within the machine precision. Furthermore, if the value of r is larger, the error
will be smaller. In addition, the larger the value of n is, the larger the error
will be. Thus, Algorithm 1 is feasible and effective to solve the GIEPPJ for a modest order n.
Table 1. Numerical results of Example 4.2.
5. Conclusions
In this paper, we have considered the generalized inverse eigenvalue problem for the class of pseudo-Jacobi matrices . The three-term recurrence relation of the leading principal minors of the matrix pencil
is central in obtaining the solution. We analysed the properties of the eigenvectors of the matrix pair
, and we obtained a necessary and sufficient condition for the solvability of the
. The given conditions ensure that the
has a unique solution. The obtained pseudo-Jacobi matrix in
was reconstructed from the prescribed mixed spectral data, namely, the given matrices
,
, two distinct real eigenvalues and their corresponding partial eigenvectors. Finally, a numerical algorithm to solve the
was provided. Some illustrative examples were also given to test the efficiency of the algorithm. The results in this paper extend the previous results obtained by Ghanbari and Parvizpour [Citation18], and Sen and Sharma [Citation19] into the non-self-adjoint case. For the class of pseudo-Jacobi matrices in
with
,
,
,
and 0<r<n−1, the GIEPPJ is an open problem under investigation.
Acknowledgements
The authors are very grateful to the anonymous referees for careful reading of the manuscript and valuable comments and suggestions which helped to improve the presentation of this paper.
Disclosure statement
No potential conflict of interest was reported by the authors.
Additional information
Funding
References
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