# Row stochastic inverse eigenvalue problem

- Yang Shang-jun
^{1}and - Xu Chang-qing
^{2}Email author

**2011**:24

https://doi.org/10.1186/1029-242X-2011-24

© Shang-jun and Chang-qing; licensee Springer. 2011

**Received: **20 January 2011

**Accepted: **21 July 2011

**Published: **21 July 2011

## Abstract

In this paper, we give sufficient conditions or realizability criteria for the existence of a row stochastic matrix with a given spectrum Λ = {*λ*_{1}, ..., *λ*_{
n
}} = Λ_{1} ∪ ⋯ ∪ Λ_{
m
} ∪ Λ_{m+1}, *m >* 0; where
(*p*_{
k
} is an integer greater than 1), *λ*_{k 1}= *λ*_{
k
} *>* 0, 1 = *λ*_{1} ≥ *ω*_{
k
} *>* 0, *k* = 1, ..., *m*; Λ_{m+1}= {*λ*_{
m
}+1}, *ω*_{m+1}≡ *λ*_{1} + ..., +*λ*_{
n
} ≤ *λ*_{1}, *ω*_{
k
} ≥ *λ*_{
k
}, *ω*_{1} ≥ *λ*_{
k
}, *k* = 2, ..., *m* + 1. In the case when *p*_{1}, ..., *p*_{
m
} are all equal to 2, Λ becomes a list of 2*m* + 1 real numbers for any positive integer *m*, and our result gives sufficient conditions for a list of 2*m* + 1 real numbers to be realizable by a row stochastic matrix.

**AMS classification:** 15A18.

## Keywords

## 1 Introduction and preliminaries

A list of complex numbers Λ is realizable by a matrix *A* if Λ is the spectrum of *A*. An *n* × *n* nonnegative matrix *A* = (*a*_{
mk
}) = is a row stochastic matrix if
, *m* = 1, ..., *n*. The row stochastic inverse eigenvalue problem is the problem of characterizing all possible spectrum of row stochastic matrices. Since row stochastic matrices are important in applications this kind of inverse eigenvalue problems should be interesting. There are also some papers that contribute to the study of the doubly stochastic inverse eigenvalue problem (e.g., [1, 2] and references therein).

In this paper, we give sufficient conditions for some lists of complex numbers, including lists of 2*m* + 1 real numbers, to be realizable by a row stochastic matrix.

We use
to denote the fact that the *n* × *n* real square matrix *A* = (*a*_{
mk
}) satisfies
, *m* = 1, ..., *n*; use *A* = diag(*A*_{1}, ..., *A*_{
t
}) to denote the fact that *A* is a block diagonal matrix with diagonal blocks *A*_{1}, ..., *A*_{
t
}; use *σ*(*A*) to denote the spectrum of *A*; use *P* (*n*) to denote the permutation matrix of order *n* the *k* th row of which is the (*k* + 1)th row of *I*_{
n
} with the first row being (1, 0, ..., 0). Later, we will make use of the fact that the spectrum of *ωP*(*n*) is {*ω*, *ωe*^{2πi/n}, *ωe*^{4πi/n}, ..., *ωe*^{(n-1)πi/n}}.

Since our results are based on the following theorem from [3], we restate it with the proof.

**Theorem 1** (Brauer extended) [3] Let *A* be an *n* × *n* arbitrary matrix with eigenvalues *λ*_{1}, ..., *λ*_{
n
} and *X* = (*x*_{1}, ..., *x*_{
t
}) be such that rank(*X*) = *t* and *Ax*_{
k
} = *λ*_{
k
}*x*_{
k
}, *k* = 1, ..., *t*, *t* ≤ *n*. Let *C* be a *t* × *n* arbitrary matrix. Then the matrix *A* + *XC* has eigenvalues *μ*_{1}, ..., *μ*_{
t
}, *λ*_{t+1}, ..., *λ*_{
n
}, where *μ*_{1}, ..., *μ*_{
t
}are eigenvalues of the matrix *D* + *CX* with *D* = diag(*λ*_{1}, ..., *λ*_{
t
}).

**Proof**Let

*S*= (

*X*,

*Y*) be a nonsingular matrix with . Then

*UX*=

*I*

_{ t },

*V Y*=

*I*

_{n-t},

*VX*=

*UY*= 0. Let

*C*=

*C*

_{1},

*C*

_{2}), , where both

*C*

_{1},

*X*

_{1}are both

*t*×

*t*and

*Y*

_{1}is

*t*× (

*n*-

*t*). since

*AX*=

*XD*, we have

◊

is a row stochastic matrix with eigenvalues *λ*_{1}, ..., *λ*_{
t
} and diagonal entries *ω*_{1}, ..., *ω*_{
t
}.

**Proof**It is clear that has diagonal entries

*ω*

_{1}, ...,

*ω*

_{ t }and is a nonnegative matrix by (2) and (4). In addition, the eigenpolynomial of

*B*is factorized as

◊

## 2 Main results

Since the set Λ in Theorem 2 is assumed to be a list of complex numbers, it could be considered as a generalization of Theorem 8 of [3] (Λ in Theorem 8 of [3] is assumed to be a real list). But the representation and the proof of these two theorems almost have no difference.

**Theorem 2** [3] Let Λ = {*λ*_{1}, ..., *λ*_{
n
}} be a list of complex numbers. If there exists a partition Λ = Λ_{1} ∪ ⋯ ∪ Λ_{
t
},
with *λ*_{11} = *λ*_{1} ≥ *λ*_{21} ≥ *λ*_{31} ≥ ⋯ ≥ *λ*_{t 1}> 0 and for each Λ_{
k
} we associate a corresponding list
, 0 ≤ *ω*_{
k
} ≤ *ω*_{1} which is realizable by a nonnegative matrix
of order *p*_{
k
}, *k* = 1, ..., *t*, as well as there exists a nonnegative matrix
of order *t*, which has eigenvalues {*λ*_{1}, *λ*_{21}, ..., *λ*_{t 1}} and diagonal entries {*ω*_{1}, ..., *ω*_{
t
}}, then Λ is realizable by an *n* × *n* nonnegative matrix
.

**Proof** Note the *p*_{
k
}-dimensional vector (1, ..., 1)^{T} is an eigenvector of
corresponding to the eigenvalue *ω*_{
k
}. Let *X* = (*X*_{1}, ..., *X*_{
t
}), where *X*_{
k
} = (0, ... 0, 1, ..., 1, 0, ..., 0)^{T} is an *n*-dimensional vector with *p*_{
k
} ones from the position *p*_{1} + ⋯ + *p*_{k-1}+ 1 to *p*_{1} + ⋯ + *p*_{
k
} and zeros elsewhere. Let *A* = diag(*A*_{1}, ..., *A*_{
t
}), *D* = diag(*ω*_{1}, ..., *ω*_{
t
}), then *X* is of rank *t* and *AX* = *XD*.

*C*= (

*C*

_{1}, ...,

*C*

_{ t }), where

*C*

_{ k }is the

*t*×

*p*

_{ k }matrix whose first column is (

*c*

_{1k},

*c*

_{2k}, ...,

*c*

_{ tk })

^{T}and whose other entries are all zero. Then

where *C*_{
mk
} is the *p*_{
m
} × *p*_{
k
} matrix whose first column is (*c*_{
mk
}, *c*_{
mk
}, ..., *c*_{
mk
})^{T} and whose other entries are all zero. Now we chose *C* with *c*_{11}, ..., *c*_{
tt
} = 0 so that the matrix
. Then for this choice of *C*, we conclude that
is nonnegative with spectrum Λ by Theorem 1. ◊

**Theorem 3**Let a list of complex numbers Λ = {

*λ*

_{1}, ...,

*λ*

_{ n }} = Λ

_{1}∪ ⋯ ∪ Λ

_{ m }∪ Λ

_{m+1},

*m >*0; (

*p*

_{ k }is an integer greater than 1),

*λ*

_{k 1}=

*λ*

_{ k }

*>*0,

*k*= 1, ...,

*m*; Λ

_{m+1}= {

*λ*

_{m+1}} be such that 1 =

*λ*

_{1}≥

*ω*

_{ k }

*>*0,

*k*= 1, 2, ...,

*m*. Let

where *M*_{
kk
} = *ω*_{
k
}*P*(*p*_{
k
}), *k* = 1, ..., *m* + 1; *M*_{
kj
} is the *p*_{
k
} × *p*_{
j
} matrix whose first column is (*ω*_{
j
} - *λ*_{
j
}, ..., *ω*_{
j
} - *λ*_{
j
})^{T}, *k* ≠ *j*, *j* = 2, ..., *m* + 1 and whose other entries are all zero; *M*_{k 1}is the *p*_{
k
} × *p*_{1} matrix whose first column is (*ω*_{1} - *λ*_{
k
}, ..., *ω*_{1} - *λ*_{
k
})^{T}, *k* = 2, ..., *m* + 1 and whose other entries are all zero.

**Proof** It is clear that
is realizable by the nonnegative matrix
, *k* = 1, ..., *m* + 1. Since
, *k* = 1, ..., *m*, we have *ω*_{m+1}= *s* = *λ*_{1} + ⋯ + *λ*_{
n
} = *λ*_{1} - *ω*_{1} + ⋯ + *λ*_{
m
} - *ω*_{
m
} + *λ*_{m+1}by Condition (6) and hence *λ*_{1} + ⋯ + *λ*_{m+1}= *ω*_{1} + ⋯ + *ω*_{m+1}. Meanwhile if (7) and (8) hold, then all conditions of Lemma 1 are satisfied and hence the row stochastic matrix *B* defined in (5) with *t* = *m* + 1 has eigenvalues {*λ*_{1}, *λ*_{2}, ..., *λ*_{m+1}} and diagonal entries {*ω*_{1}, ..., *ω*_{m+1}}. Therefore, the list Λ must be realizable by an *n* × *n* row stochastic matrix *M* by Theorem 2. Applying Theorem 2, we compute the solution matrix *M* and get the result as defined in (9). ◊

When *p*_{
k
} ≤ 2 for all *k* = 1, ..., *m* + 1, the set Λ in Theorem 3 becomes a list of real numbers. In this case, applying Theorem 3, we have the following result for real row stochastic inverse eigenvalues problem.

**Theorem 4**If a list of real numbers Λ = {

*λ*

_{1}, ...,

*λ*

_{2m+1}} = Λ

_{1}∪ ⋯ ∪Λ

_{ m }∪{

*λ*

_{m+1}}; Λ

_{ k }= {

*λ*

_{ k },

*λ*

_{2m+2-k}},

*k*= 1, ...,

*m*satisfies

**Proof** Let *μ*_{
k
} = *λ*_{
k
}, *ω*_{
k
} = -*λ*_{2m+2 - k}, *p*_{
k
} = 2, *k* = 1, ..., *m*, *ω*_{m+1}= *s*, *p*_{m+1}= 1, then all the conditions of Theorem 3 are satisfied and Λ is realizable by the row stochastic matrix *M* defined in (9) by Theorem 3. In the case of Theorem 4, the matrix in (9) becomes the matrix in (13). Therefore, Λ is realizable by the row stochastic matrix *M* in (13). ◊

**Remark** Theorem 10 of [3] gives sufficient conditions only for a list of 5 real numbers to be the spectrum of some 5 × 5 nonnegative matrix
; our Theorem 4 gives sufficient conditions for a list of 2*m* + 1 real numbers for any integer *m >* 0 to be the spectrum of some row stochastic matrix. In addition, the conditions of Theorem 4 are more easily handled.

## 3 Examples

**Example 1**Λ = {

*λ*

_{1}, ...,

*λ*

_{7}} = {1, 0.75, 0.7, 0.1, -0.75, -0.8, -0.8} satisfies Conditions (10), (11) and (12) of Theorem 4 with

*m*= 3,

*s*= 0.2, -

*λ*

_{5}-

*λ*

_{2}= -

*λ*

_{6}-

*λ*

_{3}= 0.05,

*s*-

*λ*

_{4}= 0.1, -

*λ*

_{7}-

*λ*

_{2}= 0.05, -

*λ*

_{7}-

*λ*

_{3}= 0.1, -

*λ*

_{7}-

*λ*

_{4}= 0.7. Therefore Λ is realizable by the following row stochastic matrix:

**Example 2**Λ = {

*λ*

_{1}, ...,

*λ*

_{6}} = {1, 0.2, 0.7e

^{2πi/5}, 0.7e

^{4πi/5}, 0.7e

^{6πi/5}, 0.7e

^{8πi/5}} satisfies all the conditions of Theorem 3 with

*m*= 1,

*p*

_{1}= 5,

*p*

_{2}= 1,

*ω*

_{2}=

*s*= 0.5 < 1 =

*λ*

_{1},

*ω*

_{1}= 0.7 >

*ω*

_{2}> 0.2 =

*λ*

_{2},

*ω*

_{2}-

*λ*

_{2}= 0.3,

*ω*

_{1}-

*λ*

_{2}= 0.5. Therefore, Λ is realizable by the following row stochastic matrix:

## Declarations

### Acknowledgements

We thank the anonymous referee for his/her kind suggestion that leads us to notice the results in [2]. This study was supported by the NSFchina#10871230 and Innovation Group Foundation of Anhui University #KJTD001B.

## Authors’ Affiliations

## References

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