*L*^{p}-Theory of Differential Forms and Related Operators with Applications

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Research Article | Open Access

# On the Inverse Eigenvalue Problem for Irreducible Doubly Stochastic Matrices of Small Orders

**Academic Editor:**Shusen Ding

#### Abstract

The inverse eigenvalue problem is a classical and difficult
problem in matrix theory. In the case of real spectrum, we first present some sufficient
conditions of a real *r*-tuple (for ; 3; 4; 5) to be realized by a
symmetric stochastic matrix. Part of these conditions is also extended
to the complex case in the case of complex spectrum where the realization matrix may not
necessarily be symmetry. The main approach throughout the paper in
our discussion is the specific construction of realization matrices and the
recursion when the targeted *r*-tuple is updated to a -tuple.

#### 1. Introduction

For a square matrix , let denote the spectrum of . Given an -tuple of numbers, real or complex, the problem of deciding the existence of a nonnegative matrix with is called the nonnegative inverse eigenvalue problem (NIEP) which has for a long time been one of the problems of main interest in the theory of matrices.

Sufficient conditions for the existence of an entrywise positive matrix with have been investigated by many authors [1–14]. The case is trivial. The problem has been solved for by Loewy and London [6]. The cases and have been solved for matrices with trace zero by Reams [10] and Laffey and Meehan [5], respectively. So, for real spectra, complete constructive solutions to NIEP are available for . For the case of nonreal spectra for , complete solutions are available through the work of Laffey and Meehan [5], independently, and that of Torre-Mayo et al. [12] by analyzing coefficients of the characteristic polynomial. EBL digraphs, and for , complete solutions are available through the work of Nazari and Sherafat [8].

An nonnegative matrix is called a row stochastic matrix if , ; is called a doubly stochastic matrix if , , and , . Since row stochastic and doubly stochastic matrices are important nonnegative matrices, it is surely important to investigate the existence of row or doubly stochastic matrices with prescribed spectrum under certain conditions. We call this special NIEP the row or doubly stochastic inverse eigenvalue problem (RSIEP or DSIEP). Hwang and Pyo [3] gave some interesting results for the symmetric DSIEP.

An -tuple is *nonnegative (doubly stochastic) realizable* if there exists an nonnegative (doubly stochastic) matrix with . In this case, we say is a *nonnegative (doubly stochastic) realization* of or *the nonnegative (doubly stochastic) matrix ** that realizes the **-tuple *.

A nonincreasing -tuple is called -feasible if it satisfies(1);(2) for all .

Throughout the paper, we denote the spectrum of by ; the spectrum radius of by ; the all-ones column vector of dimensions by . We use for the dimensional normalized vector and for the identity matrix of order .

Theorem NN (see [15]). *Let ** be an irreducible nonnegative matrix of order **. Then, we have *(a)* and *;(b)* for some** in**, and the null space of ** is of dimension 1;*(c)* for any *.

In this paper, we study DSIEP of order . In Section 2, we present some sufficient conditions for the DSIEP for a given real -tuple. In Section 3, we present some sufficient conditions for the DSIEP for a given nonreal complex -tuple where the realization matrix may not necessarily be symmetric.

#### 2. The Case of Real Spectrum

Lemma 1. *If , then there is a irreducible doubly stochastic matrix realizing .*

*Proof. *It is easy to verify that the following irreducible doubly stochastic matrix:
realizes .

In this section, we present a theorem that is analogy to Theorem 2.1 of [8]. The theorem is used to construct an irreducible symmetric stochastic realization of a given -tuple with designed conditions.

Theorem 2. *For any integer if is an irreducible doubly stochastic matrix with and , then there exists an irreducible doubly stochastic matrix such that , where ; .*

*Proof. *We know, by Theorem NN, that is a simple eigenvalue of and is the unique normalized positive eigenvector associated with 1 such that , , . Now we can find an matrix such that is a unitary matrix. Then,
where , and it is not necessary to know the value of each entry in the location remarked by “*.” Since , we have . By Schur’s unitary triangularization theorem (see [15]), there exists a unitary matrix of order , such that , where is a upper triangular matrix and is the set of all diagonal entries of . Now for the unitary matrix , we have
where is unitary with and
Therefore,
where by (5) and

It is easy to verify that the following matrix:
is a doubly stochastic matrix. Let ; then the following matrix:
is a unitary matrix, since
by , , and . In addition using and , we have
where
Therefore, . Since , we have , and hence . In addition, (for is a doubly stochastic matrix) implies that . So the spectrum of is . Now a direct calculation produces the following:
Finally, the irreducible doubly stochastic matrix has the desired spectrum .

The following result is obtained by a similar argument used in Theorem 2.

Corollary 3. *Let be an irreducible row (symmetric) stochastic matrix with -tuple as its spectrum. Denote , where . Then can be realized by an irreducible row (symmetric) stochastic matrix.*

Notice that a real -tuple is realized by an irreducible doubly stochastic matrix only if is a simple eigenvalue of . For convenience, we always assume that .

Corollary 4. *If the real triple satisfies
**
then is realized by a symmetric irreducible stochastic matrix.*

*Proof. *Assume that satisfies Condition (13) which implies that . Let , ; then , . Let and
Then is an irreducible doubly stochastic matrices with by Lemma 1 and
is nonnegative. Finally, the matrix of order in (7)
is irreducible symmetric stochastic with by Theorem 2.

*Remark 5. *Theorem 14 of [8] shows that Condition (13) is sufficient and necessary for a real triple to be doubly stochastic realizable.

Corollary 6. *If a real feasible nonincreasing 4-tuple satisfies
**
then is realized by a symmetric irreducible stochastic matrix.*

*Proof. *Assume that satisfies Condition (17). Then yields . Let , ; then , . It is clear that since would produce , which, together with (17), yields
It follows that , which is a contradiction. So . Since by (17), the following matrix:
is irreducible doubly stochastic by whom is realized by Corollary 4 and
is nonnegative. Finally, the matrix of order in (7)

is irreducible symmetric stochastic with by Theorem 2.

Using this recursive method, we can prove the following result.

Corollary 7. *Let be -feasible and satisfies
**
then can be realized by a symmetric irreducible stochastic matrix.*

*Proof. *Assume that satisfies Condition (22) which implies that . Let ; then . Now under Condition (22) using the same recursive method and Theorem 2, we can construct the following irreducible symmetric stochastic matrix:

by whom is realized.

The next result shows that under some stronger conditions, we can construct a nonsymmetric irreducible doubly stochastic matrix to realize the given real triple.

Proposition 8. *Let satisfy . If
**
then is realized by a nonsymmetric irreducible doubly stochastic matrix.*

*Proof. *If satisfies Condition (24), then construct the following bunch of irreducible matrices with one parameter and trace of whose row sums are all equal to 1:
Then and become which has a positive zero as follows:
Moreover,
In addition, since
we have
Therefore, is an irreducible doubly stochastic matrix with and hence .

*Remark 9. *Corollary 6 produces the sufficient condition (13) for an irreducible symmetric stochastic matrix of order 3 to have the prescribed real spectrum, and Proposition 8 produces the sufficient condition (24) for an irreducible nonsymmetric doubly stochastic matrix of order 3 to have the prescribed real spectrum. Note that Condition (24) implies Condition (13) because if .

Theorem M (see [7]). *Let with . Then is realized by a symmetric doubly stochastic matrix with zero trace if and only if , and when satisfies the condition, the matrix is
*

Corollary 10. *Let and . If
**
then is realized by a symmetric doubly stochastic matrix with zero trace.*

*Proof. *Let . If (32) holds, then we have and . So is realized by the doubly stochastic matrix given in (31) by Theorem M. Let ; then , . Now is realized by the matrix (given in (7)) as follows:
that is an irreducible doubly stochastic with zero by Theorem 2.

#### 3. The Case of Complex Spectrum

Given a circulant doubly stochastic matrix, it is easy to obtain its spectrum (see Lemma 11). In this section, we use this result to construct an IDS (irreducible doubly stochastic) matrix to realize a given complex triple containing a pair of conjugate complex numbers with some additional conditions. This matrix is used together with Theorem 2 to construct an IDS matrix to realize a given complex 4-tuple and a 5-tuple containing exactly a pair of conjugate complex numbers with special conditions in a recursive method. Also constructed is an IDS realization of a given complex 5-tuple, which contains two pairs of conjugate complex numbers with special conditions.

The following result is well known. We give a short proof for completeness.

Lemma 11. *In the complex plane, let be the regular polygon whose vertices are all the th roots of unity as follows: , , , and let be a nonreal number. If , or equivalently, is a convex combination of the th roots of unity; that is, , , , ; then there is a doubly stochastic matrix such that
*

*Proof. *It is clear that the following permutation matrix:
has spectrum and then the following circulant matrix:
is doubly stochastic and has spectrum . When , we have .

Theorem 12. *Let with and . Then can be realized by an IDS matrix if and only if
**
When (37) holds to be realized by the irreducible doubly stochastic matrix,
**
where
*

*Proof. *Assume that (37) holds. Then, in the complex plane, is inside the regular triangle whose vertices are all the 3rd roots of unit , , and , and hence is a convex combination of , and . It is not difficult to calculate
where is given by (39). Therefore the spectrum of the irreducible doubly stochastic matrix given in (38) contains by Lemma 11. Since is a doubly stochastic matrix, we have and hence the sufficiency is proved. To prove the necessity, assume that is realized by a doubly stochastic matrix , . Then from which follows , and the sum of products of pairs eigenvalues of is
from which follows . Therefore, (37) holds.

*Remark 13. *The necessary and sufficient condition (37) for the DSIEP was given by Theorems 12 and 14 of [9].

Using Theorems 2 and 12, we have the following corollaries.

Corollary 14. *Let , , , and . If
**
then can be realized by a irreducible stochastic matrix.*

*Proof. *Let , ; then , . If (42) holds, then is realized by the irreducible doubly stochastic matrix
by Theorem 12, where
Now, the matrix (given in (7)) as follows:
is an irreducible doubly stochastic matrix whom is realized (Theorem 2).

Corollary 15. *Let be a complex 5-tuple with , , and , , , then , . If
**
then is realized by a irreducible stochastic matrix.*

*Proof. *Let , ; then , and hence . If satisfies Condition (46), then is realized by an irreducible doubly stochastic matrix by Corollary 14. Now for , the matrix (given in (7))
is an irreducible doubly stochastic by whom , , is realized by Theorem 2.

Theorem 16. *Let , contain a pair of conjugate complex numbers such that . If
**
then is realized by a irreducible doubly stochastic matrix with zero trace.*

* Proof. *Assume that (48) holds. Then, in the complex plane, is inside the right triangle whose vertices are , , and hence is a convex combination of , and . It is not difficult to calculate
Since , , , are all the 4th roots of unit, Lemma 11 asserts that the spectrum of the following irreducible doubly stochastic matrix:
is , , .

Corollary 17. *Let and . If
**
then is a realized by a irreducible stochastic matrix.*

*Proof. *Let , ; then . If (51) holds, then is realized by the irreducible doubly stochastic matrixby Theorem 16. Now the following matrix (given in (7)):
is an irreducible doubly stochastic matrix with zero trace by whom is realized by Theorem 2.

Theorem 18. *Let contain two pairs of conjugate complex numbers with , given and , depending on (). If
**
then is realized by a irreducible doubly stochastic matrix, where , are depending on (see (58) and (63)).*

*Proof. *Assume that (54) holds. Then, in the complex plane, is inside the right pentagon whose vertices are all the 5th roots of units , , (see Figure 1) and hence is a convex combination of . There are two cases to be considered.*Case 1. * satisfy (54) and . In this case, is inside Triangle and hence is a convex combination of . A calculation yields
where
Now the following irreducible circulant matrix that is also doubly stochastic:
has spectrum: by Lemma 11. Taking , we have and taking , we have that and are in and are conjugate to each other. Therefore, if we set
then is realized by the irreducible doubly stochastic matrix .*Case 2. * satisfy (54) and . In this case, is inside Triangle and hence is a convex combination of . A calculation yields
where
with
Now the irreducible circulant matrix that is also doubly stochastic as follows:
has spectrum by Lemma 11. Taking , we have and taking , we have that and are in and are conjugate to each other. Therefore, if we set
then is realized by the irreducible doubly stochastic matrix .

*Example 19. * satisfies Condition (37) of Theorem 12 and is doubly stochastic realized by

*Example 20. *Let , . Then Condition (42) of Corollary 14 is satisfied and is doubly stochastic realized by

*Example 21. *Let . Then