Construction of 4 x 4 symmetric stochastic matrices with given spectra

Jaewon Jung; Donggyun Kim

doi:10.1515/math-2023-0176

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Construction of 4 x 4 symmetric stochastic matrices with given spectra

Jaewon Jung and Donggyun Kim

Published/Copyright: March 16, 2024

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$Open Mathematics$

From the journal Open Mathematics Volume 22 Issue 1

Abstract

The symmetric stochastic inverse eigenvalue problem (SSIEP) asks which lists of real numbers occur as the spectra of symmetric stochastic matrices. When the cardinality of a list is 4, Kaddoura and Mourad provided a sufficient condition for SSIEP by a mapping and convexity technique. They also conjectured that the sufficient condition is the necessary condition. This study presents the same sufficient condition for SSIEP, but we do it in terms of the list elements. In this way, we provide a different but more straightforward construction of symmetric stochastic matrices for SSIEP compared to those of Kaddoura and Mourad.

Keywords: nonnegative inverse eigenvalue problem; symmetric matrices; symmetric stochastic matrices; symmetric stochastic inverse eigenvalue problem

MSC 2010: 15A18; 15A51

1 Introduction

The nonnegative inverse eigenvalue problem asks which lists of complex numbers occur as the spectra of nonnegative matrices. This is a long-standing problem in matrix theory (for example, see a survey paper [1]). The nonnegative matrices could be specified as, for example, symmetric, stochastic, doubly stochastic, symmetric stochastic, or nonspecified forms.

A real square matrix with nonnegative entries is said to be (generalized) symmetric doubly stochastic or simply (generalized) symmetric stochastic if it is symmetric and all of its row or column sums are equal to a nonnegative constant α . We will call it a symmetric stochastic matrix simply. The constant α could be any nonnegative number throughout this study, including the usual case α = 1 . In this way, we can trace the behavior of α ; otherwise, it is concealed when α = 1 .

The nonnegative inverse eigenvalue problem for a given list where the form of the matrix is symmetric stochastic is called a symmetric stochastic inverse eigenvalue problem or SSIEP.

Let Λ be a list of real numbers and n the cardinality of Λ . When n is 1 or 2, the SSIEP is easy. This SSIEP has only been solved for the case n = 3 by Perfect and Mirsky [2] in 1965. For the case n ≥ 5 , the SSIEP is wildly open (see, for example, [3–5]).

For the case n = 4 , a sufficient condition for SSIEP is given in [2], and Mourad and coauthors provided a sufficient condition that covered a more comprehensive range for SSIEPs by a mapping and convexity technique and conjectured that the sufficient condition is the necessary condition in [6–8].

This study presents a different but more straightforward construction of symmetric stochastic matrices for SSIEP when n = 4 , compared to those of Mourad in [6]. A particular orthogonal matrix (1) is used, and the symmetric stochastic matrices are expressed simply in terms of the list elements Λ (see below A 1 , A 2 , and A 3 ), and we arrived at the same conjecture in [8].

2 Symmetric stochastic matrices

Let a matrix U be of the form:

(1) U = 1 2 1 2 1 2 1 2 − 1 2 − 1 2 1 2 1 2 − a a − b b − b b a − a .

This matrix U becomes orthogonal if and only if the numbers a and b satisfy the relation a 2 + b 2 = 1 2 , because the product of the transpose of U and U is of the form:

U t U = 1 2 + a 2 + b 2 1 2 − a 2 − b 2 0 0 1 2 − a 2 − b 2 1 2 + a 2 + b 2 0 0 0 0 1 2 + a 2 + b 2 1 2 − a 2 − b 2 0 0 1 2 − a 2 − b 2 1 2 + a 2 + b 2 .

Let Λ = { λ 1 , λ 2 , λ 3 , λ 4 } be a list of real numbers with nonincreasing order and [ Λ ] the diagonal matrix with the diagonal entries Λ . The product U t [ Λ ] U becomes a symmetric matrix with each row and column summing to λ 1 :

(2) U t [ Λ ] U = λ 1 4 + λ 2 4 + a 2 λ 3 + b 2 λ 4 λ 1 4 + λ 2 4 − a 2 λ 3 − b 2 λ 4 λ 1 4 − λ 2 4 + a b λ 3 − a b λ 4 λ 1 4 − λ 2 4 − a b λ 3 + a b λ 4 λ 1 4 + λ 2 4 − a 2 λ 3 − b 2 λ 4 λ 1 4 + λ 2 4 + a 2 λ 3 + b 2 λ 4 λ 1 4 − λ 2 4 − a b λ 3 + a b λ 4 λ 1 4 − λ 2 4 + a b λ 3 − a b λ 4 λ 1 4 − λ 2 4 + a b λ 3 − a b λ 4 λ 1 4 − λ 2 4 − a b λ 3 + a b λ 4 λ 1 4 + λ 2 4 + b 2 λ 3 + a 2 λ 4 λ 1 4 + λ 2 4 − b 2 λ 3 − a 2 λ 4 λ 1 4 − λ 2 4 − a b λ 3 + a b λ 4 λ 1 4 − λ 2 4 + a b λ 3 − a b λ 4 λ 1 4 + λ 2 4 − b 2 λ 3 − a 2 λ 4 λ 1 4 + λ 2 4 + b 2 λ 3 + a 2 λ 4 .

If the matrix U is orthogonal, then the symmetric matrix U t [ Λ ] U has the eigenvalues Λ . Furthermore, if all entries of the matrix U t [ Λ ] U are nonnegative, the matrix U t [ Λ ] U becomes a symmetric stochastic matrix with each row and column summing to λ 1 , whose spectrum is the list Λ .

The following theorem presents a sufficient condition for the SSIEP, which Kaddoura and Mourad proved in [8]. Still, we give a different proof where the particular orthogonal matrix (1) is used, and so different and simple types of symmetric stochastic matrices are obtained. The symmetric stochastic matrices are expressed in terms of the list Λ .

Theorem 2.1

Let Λ = { λ 1 , λ 2 , λ 3 , λ 4 } be a list of real numbers with the following conditions:

(3) λ 1 ≥ λ 2 ≥ λ 3 ≥ λ 4 ≥ − λ 1 ,

(4) λ 1 + λ 2 + λ 3 + λ 4 ≥ 0 ,

(5) ( λ 1 + λ 3 ) ( λ 1 + λ 4 ) + ( λ 2 + λ 3 ) ( λ 2 + λ 4 ) ≥ 0 .

Then, there is a symmetric stochastic matrix whose spectrum is the list Λ .

Proof

Let R be the region consisting of lists ( λ 1 , λ 2 , λ 3 , λ 4 ) , which satisfy Conditions (3)–(5). We split the region R into three subregions.

Let R 1 be the subregion of R whose lists satisfy the further conditions:

(6) λ 4 < 0 , λ 3 < − λ 4 , λ 2 < − λ 3 + λ 4 2 , λ 1 ≥ λ 2 + λ 3 − λ 4 ,

R 2 the subregion of R whose lists satisfy the further conditions:

(7) λ 4 < 0 , λ 3 < − λ 4 , λ 1 < λ 2 + λ 3 − λ 4 , λ 1 < − λ 2 − 2 λ 4 ,

and R 3 the subregion of R whose lists satisfy the further conditions:

(8) λ 4 < 0 , λ 3 < − λ 4 , λ 2 ≥ − λ 3 + λ 4 2 , λ 1 ≥ − λ 2 − 2 λ 4 , or

(9) λ 4 < 0 , λ 3 ≥ − λ 4 , or

(10) λ 4 ≥ 0 .

First, check that the union of the subregions R 1 , R 2 , and R 3 , is the region R . Let λ = ( λ 1 , λ 2 , λ 3 , λ 4 ) be an arbitrary list in R . We will show that λ is placed in one of the subregions. If λ 4 ≥ 0 , then λ ∈ R 3 by the definition of R 3 . Now, assume that λ 4 < 0 . If λ 3 ≥ − λ 4 , then λ ∈ R 3 by the definition of R 3 . Now, assume further that λ 3 < − λ 4 . If λ 1 < λ 2 + λ 3 − λ 4 and λ 1 < − λ 2 − 2 λ 4 , then by definition of R 2 , λ ∈ R 2 .

Now, consider the remaining region that λ 1 ≥ λ 2 + λ 3 − λ 4 or λ 1 ≥ − λ 2 − 2 λ 4 . If we assume further that 2 λ 2 < − λ 3 − λ 4 , then

− λ 2 − 2 λ 4 = ( − λ 2 − 2 λ 4 ) − ( λ 2 + λ 3 − λ 4 ) + ( λ 2 + λ 3 − λ 4 ) = − ( 2 λ 2 + λ 3 + λ 4 ) + ( λ 2 + λ 3 − λ 4 ) > λ 2 + λ 3 − λ 4 .

Therefore, this region can be written as λ 1 ≥ λ 2 + λ 3 − λ 4 and 2 λ 2 < − λ 3 − λ 4 . Hence, we have that λ ∈ R 1 . Now, we assume the remaining case that 2 λ 2 ≥ − λ 3 − λ 4 , then

λ 2 + λ 3 − λ 4 = ( λ 2 + λ 3 − λ 4 ) − ( − λ 2 − 2 λ 4 ) + ( − λ 2 − 2 λ 4 ) = ( 2 λ 2 + λ 3 + λ 4 ) + ( − λ 2 − 2 λ 4 ) ≥ − λ 2 − 2 λ 4 .

Therefore, this region can be written as λ 1 ≥ − λ 2 − 2 λ 4 and 2 λ 2 ≥ − λ 3 − λ 4 . Hence, we have that λ ∈ R 3 . This completes the proof that the union of the subregions R 1 , R 2 , and R 3 is the region R .

Let us consider the lists in the subregion R 1 . In this subregion R 1 , set the matrix U with a = 1 2 and b = 1 2 in (1), and then, U becomes orthogonal. Let a list Λ be in R 1 and a matrix A 1 the product U t [ Λ ] U in (2), where [ Λ ] is the diagonal matrix with diagonal entries Λ . The matrix A 1 is of the form:

A 1 = 1 4 λ 1 + λ 2 + λ 3 + λ 4 λ 1 + λ 2 − λ 3 − λ 4 λ 1 − λ 2 + λ 3 − λ 4 λ 1 − λ 2 − λ 3 + λ 4 λ 1 + λ 2 − λ 3 − λ 4 λ 1 + λ 2 + λ 3 + λ 4 λ 1 − λ 2 − λ 3 + λ 4 λ 1 − λ 2 + λ 3 − λ 4 λ 1 − λ 2 + λ 3 − λ 4 λ 1 − λ 2 − λ 3 + λ 4 λ 1 + λ 2 + λ 3 + λ 4 λ 1 + λ 2 − λ 3 − λ 4 λ 1 − λ 2 − λ 3 + λ 4 λ 1 − λ 2 + λ 3 − λ 4 λ 1 + λ 2 − λ 3 − λ 4 λ 1 + λ 2 + λ 3 + λ 4 .

Then, the matrix A 1 is symmetric with each row and column summing to λ 1 , and its spectrum is Λ . If we show that each entry of A 1 is nonnegative, then A 1 is a symmetric stochastic matrix with the spectrum Λ , and therefore, the proof is completed for the lists in the subregion R 1 .

We denote the ( i , j ) -th entry of A 1 as ( A 1 ) i j . By Condition (4), we see that 4 ( A 1 ) 11 = λ 1 + λ 2 + λ 3 + λ 4 ≥ 0 . By Condition (3), 4 ( A 1 ) 12 = ( λ 1 − λ 3 ) + ( λ 2 − λ 4 ) ≥ 0 and 4 ( A 1 ) 13 = ( λ 1 − λ 2 ) + ( λ 3 − λ 4 ) ≥ 0 . By the fourth expression of Condition (6), 4 ( A 1 ) 14 = λ 1 − ( λ 2 + λ 3 − λ 4 ) ≥ 0 . All other entries of A 1 are one of the aforementioned forms, and so, they are nonnegative.

Let us consider the lists in the subregion R 2 . In this subregion R 2 , set the matrix U with a = 1 2 λ 1 + λ 2 + 2 λ 4 − λ 3 + λ 4 and b = 1 2 λ 1 + λ 2 + 2 λ 3 λ 3 − λ 4 in (1), and then, U becomes orthogonal, because a 2 + b 2 = 1 2 . Let a list Λ be in R 2 and a matrix A 2 the product U t [ Λ ] U in (2). The matrix A 2 is of the form:

A 2 = 1 4 0 2 ( λ 1 + λ 2 ) λ 1 − λ 2 + λ λ 1 − λ 2 − λ 2 ( λ 1 + λ 2 ) 0 λ 1 − λ 2 − λ λ 1 − λ 2 + λ λ 1 − λ 2 + λ λ 1 − λ 2 − λ 2 ( λ 1 + λ 2 + λ 3 + λ 4 ) 2 ( − λ 3 − λ 4 ) λ 1 − λ 2 − λ λ 1 − λ 2 + λ 2 ( − λ 3 − λ 4 ) 2 ( λ 1 + λ 2 + λ 3 + λ 4 ) ,

where λ = − ( λ 1 + λ 2 + 2 λ 3 ) ( λ 1 + λ 2 + 2 λ 4 ) . Then, the matrix A 2 is symmetric with each row and column summing to λ 1 and its spectrum is Λ . If we show that each entry of A 2 is nonnegative, then A 2 is a symmetric stochastic matrix with the spectrum Λ , and therefore, the proof is completed for the lists in the subregion R 2 .

Consider the expression λ . The first factor of λ , λ 1 + λ 2 + 2 λ 3 , is positive by Condition (4) and the second factor of λ , λ 1 + λ 2 + 2 λ 4 , is negative by the fourth expression of Condition (7); hence, we obtain that λ ≥ 0 . Therefore, 4 ( A 2 ) 13 = ( λ 1 − λ 2 ) + λ ≥ 0 . The following equality holds

1 2 ( ( λ 1 − λ 2 ) 2 − λ 2 ) = ( λ 1 + λ 3 ) ( λ 1 + λ 4 ) + ( λ 2 + λ 3 ) ( λ 2 + λ 4 ) .

Using Condition (5), we obtain that 4 ( A 2 ) 14 = ( λ 1 − λ 2 ) − λ ≥ 0 .

By Condition (3), we see that 2 ( A 2 ) 12 = λ 1 + λ 2 ≥ 0 . By Condition (4), 4 ( A 2 ) 33 = λ 1 + λ 2 + λ 3 + λ 4 ≥ 0 and 4 ( A 1 ) 13 = ( λ 1 − λ 2 ) + ( λ 3 − λ 4 ) ≥ 0 . By the second expression of Condition (7), 2 ( A 2 ) 34 = − λ 3 − λ 4 ≥ 0 . All other entries of A 2 are one of the aforementioned forms, and so, they are nonnegative.

Finally consider the lists in the subregion R 3 . In this subregion R 3 , set the matrix U with a = 0 and b = 1 2 in (1), and then, U becomes orthogonal. Let a list Λ be in R 3 and a matrix A 3 the product U t [ Λ ] U in (2). The matrix A 3 is of the form:

A 3 = 1 4 λ 1 + λ 2 + 2 λ 4 λ 1 + λ 2 − 2 λ 4 λ 1 − λ 2 λ 1 − λ 2 λ 1 + λ 2 − 2 λ 4 λ 1 + λ 2 + 2 λ 4 λ 1 − λ 2 λ 1 − λ 2 λ 1 − λ 2 λ 1 − λ 2 λ 1 + λ 2 + 2 λ 3 λ 1 + λ 2 − 2 λ 3 λ 1 − λ 2 λ 1 − λ 2 λ 1 + λ 2 − 2 λ 3 λ 1 + λ 2 + 2 λ 3 .

Then, the matrix A 3 is symmetric with each row and column summing to λ 1 and its spectrum is Λ . If we show that each entry of A 3 is nonnegative, then A 3 is a symmetric stochastic matrix with the spectrum Λ , and therefore, the proof is completed for the lists in the subregion R 3 .

Consider the entry 4 ( A 3 ) 11 = λ 1 + λ 2 + 2 λ 4 . If the list Λ in R 3 satisfies Condition (8), by the fourth expression, we obtain that ( A 3 ) 11 ≥ 0 . If the list Λ in R 3 satisfies Condition (9), by the second expression of Condition (9), we compute that λ 1 + λ 2 + 2 λ 4 ≥ λ 1 + λ 2 − 2 λ 3 = ( λ 1 − λ 3 ) + ( λ 2 − λ 3 ) ≥ 0 ; therefore, we obtain that ( A 3 ) 11 ≥ 0 . If the list Λ in R 3 satisfies Condition (10), immediately, we obtain that ( A 3 ) 11 ≥ 0 . It is easy to check that all other entries of A 3 are nonnegative using Conditions (3)–(5).

We have shown that for every list Λ in the region R , there is a symmetric stochastic matrix that depends on the conditions of Λ , whose spectrum is Λ . Therefore, the proof of the statement is completed.□

We draw a graph to picture regions R , R 1 , R 2 , and R 3 . For the values λ 4 and λ 3 fixed as λ 4 < λ 3 < 0 , we present a graph that is a cross-section of regions on the λ 2 - and λ 1 -axes plane (Figure 1).

$Figure 1 Cross-section of region R R about fixed negative values λ 4 {\lambda }_{4} and λ 3 {\lambda }_{3} .$

Figure 1

Cross-section of region R about fixed negative values λ 4 and λ 3 .

The subregion R 1 is the region bounded by lines λ 2 = λ 3 , λ 1 = − λ 2 − λ 3 − λ 4 , λ 1 = λ 2 + λ 3 − λ 4 and λ 2 = − λ 3 − λ 4 2 . The subregion R 2 is the upper part of the two regions bounded by lines λ 1 = λ 2 + λ 3 − λ 4 , and λ 1 = − λ 2 − 2 λ 4 and the circle

( λ 1 + λ 3 ) ( λ 1 + λ 4 ) + ( λ 2 + λ 3 ) ( λ 2 + λ 4 ) = 0 .

The subregion R 3 is the region bounded by lines λ 2 = − λ 3 − λ 4 2 , λ 1 = − λ 2 − 2 λ 4 , and λ 1 = λ 2 . The region R is the union of subregions R 1 , R 2 , and R 3 , which is the colored region in Figure 1.

For the values λ 1 and λ 2 fixed as λ 1 > λ 2 > 0 , we present a graph that is a cross-section of regions on the λ 4 - and λ 3 -axes plane (Figure 2). The region R is the union of subregions R 1 , R 2 , and R 3 , which is the colored region in Figure 2.

$Figure 2 Cross-section of region R R about fixed positive values λ 1 {\lambda }_{1} and λ 2 {\lambda }_{2} .$

Figure 2

Cross-section of region R about fixed positive values λ 1 and λ 2 .

For the other cases λ 4 < 0 < λ 3 or 0 < λ 4 < λ 3 , we can draw a similar but simpler graph of regions R , R 1 , R 2 , and R 3 compared to Figures 1 and 2.

Our empirical study shows that the converse statement of Theorem 2.1 is also true, i.e., for a list of real numbers Λ = { λ 1 , λ 2 , λ 3 , λ 4 } , conditions (3)–(5) are the necessary and sufficient conditions to have a symmetric stochastic matrix with the spectrum Λ . We leave it as a conjecture. This conjecture is also presented in [8].

Conjecture 2.2

The converse statement of Theorem 2.1 is true.

3 Symmetric and symmetric stochastic matrices

The symmetric nonnegative matrix case is well known by Fiedler in 1974 [9] and Soules in 1983 [10].

Theorem 3.1

Let Λ = { λ 1 , λ 2 , λ 3 , λ 4 } be a list of real numbers with the following conditions:

(3) λ 1 ≥ λ 2 ≥ λ 3 ≥ λ 4 ≥ − λ 1 and

(4) λ 1 + λ 2 + λ 3 + λ 4 ≥ 0 .

Then, there is a symmetric nonnegative matrix whose spectrum is the list Λ .

We compare the sufficient conditions to have a symmetric stochastic matrix in Theorem 2.1 and a symmetric nonnegative matrix in Theorem 3.1. If a list Λ satisfies Conditions (3) and (4) but does not satisfy Condition (5) in Theorem 2.1, then by Theorem 3.1, there is a symmetric nonnegative matrix whose spectrum is the list Λ . We provide a symmetric nonnegative matrix with spectrum Λ , which is similar to the matrix that appeared in the proof of Theorem 2.1.

Theorem 3.2

Let Λ = { λ 1 , λ 2 , λ 3 , λ 4 } be a list of real numbers with the following conditions:

(3) λ 1 ≥ λ 2 ≥ λ 3 ≥ λ 4 ≥ − λ 1 ,

(4) λ 1 + λ 2 + λ 3 + λ 4 ≥ 0 , and

(5) ( λ 1 + λ 3 ) ( λ 1 + λ 4 ) + ( λ 2 + λ 3 ) ( λ 2 + λ 4 ) < 0 .

Then, there is a symmetric nonnegative matrix whose spectrum is the list Λ .

Proof

Let a matrix V be of the form:

V = 1 2 1 2 0 0 0 0 − 1 2 − 1 2 0 0 − 1 2 1 2 1 2 − 1 2 0 0 .

Then, the matrix V is orthogonal. Suppose that a list Λ satisfies Conditions (3)–(5). Let a matrix A be the product V t [ Λ ] V . Then, the matrix A is of the form:

A = 1 2 λ 1 + λ 4 λ 1 − λ 4 0 0 λ 1 − λ 4 λ 1 + λ 4 0 0 0 0 λ 2 + λ 3 λ 2 − λ 3 0 0 λ 2 − λ 3 λ 2 + λ 3 .

Since the matrix V is orthogonal, the symmetric matrix A has the eigenvalues Λ . If we show that all entries of the matrix A are nonnegative, the matrix A becomes a symmetric nonnegative matrix whose spectrum is the list Λ .

We check easily that by Conditions (3) and (4), the entries λ 1 + λ 4 , λ 1 − λ 4 , and λ 2 − λ 3 are nonnegative. Now, check the nonnegativity of the remaining entry λ 2 + λ 3 .

Applying the expression − λ 1 ≤ λ 4 in Condition (3) to Condition (5), we obtain that

0 > ( λ 1 + λ 3 ) ( λ 1 + λ 4 ) + ( λ 2 + λ 3 ) ( λ 2 + λ 4 ) ≥ ( − λ 4 + λ 3 ) ( − λ 4 + λ 4 ) + ( λ 2 + λ 3 ) ( λ 2 + λ 4 ) = ( λ 2 + λ 3 ) ( λ 2 + λ 4 ) .

Hence, λ 2 satisfies the relation − λ 3 < λ 2 < − λ 4 , and so, the inequality λ 2 + λ 3 > 0 holds, which completes the proof of the theorem.□

When we combine the proofs of Theorems 2.1 and 3.2, we obtain another proof of Theorem 3.1.

Funding information: This research was funded by Korea University Grant.
Author contributions: All authors contributed equally to this work and have read and agreed to the published version of the manuscript.
Conflict of interest: The authors state no conflict of interest.

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Received: 2023-08-01

Revised: 2024-01-03

Accepted: 2024-01-04

Published Online: 2024-03-16

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https://doi.org/10.1515/math-2023-0176

Keywords for this article

nonnegative inverse eigenvalue problem; symmetric matrices; symmetric stochastic matrices; symmetric stochastic inverse eigenvalue problem

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