A Geršgorin-type eigenvalue localization set with n parameters for stochastic matrices

Abstract: A set in the complex plane which involves n parameters in [0, 1] is given to localize all eigenvalues di erent from 1 for stochastic matrices. As an application of this set, an upper bound for the moduli of the subdominant eigenvalues of a stochastic matrix is obtained. Lastly, we x n parameters in [0, 1] to give a new set including all eigenvalues di erent from 1, which is tighter than those provided by Shen et al. (Linear Algebra Appl. 447 (2014) 74-87) and Li et al. (Linear and Multilinear Algebra 63(11) (2015) 2159-2170) for estimating the moduli of subdominant eigenvalues.

Since the subdominant eigenvalue of a stochastic matrix is crucial for bounding the convergence rate of stochastic processes [8,[11][12][13][14], it is interesting to give a set to localize all eigenvalues di erent from , or an upper bound for the moduli of its subdominant eigenvalue [8,15].
One can use the well-known Geršgorin circle set [16] to localize all eigenvalues for a stochastic matrix. However, this set always includes the trival eigenvalue , and thus it is not always precise for capturing all eigenvalues di erent from of a stochastic matrix. Therefore, several authors have tried to modify the Geršgorin circle set to localize more precisely all eigenvalues di erent from . In [8], Cvetković et al. gave the following set. However, the set provided by Theorem 1.1 is not e ective in some cases, such as, for the class of stochastic matrices SM = {A ∈ R n×n ∶ A is stochastic, and a ii = l i = , for each i ∈ N}, for more details, see [15]. To overcome this drawback, Li and Li [15] provided another set as follows.
Recently, by taking respectively to modify the Geršgorin circle set, Shen et al. [12], and Li et al. [11] gave three sets to localize all eigenvalues di erent from . 11,12]). Let A = [a ij ] ∈ R n×n be a stochastic matrix. If λ ∈ σ(A) { }, then and Remark here that Shen et al. [12] used these three sets to localize any real eigenvalue di erent from , which are generalized to localize all eigenvalues di erent from by Li et al. [11]. Also in [11], Li and

A Geršgorin-type eigenvalue localization set with n parameters
We rst begin with an important lemma, which is used to give some modi cations of the Geršgorin circle set. Lemma 2.1 shows that once an eigenvalue localization set for B = ed T − A is given, we can get a set to localize all eigenvalues di erent from for the stochastic matrix A [11]. Now we present the following choice of d: where By Lemma 2.1 and (1), we can obtain the following set to localize all eigenvalues di erent from of a stochastic matrix.
By applying the Geršgorin circle theorem to B α i , we have that for anyλ ∈ σ(B α i ), Furthermore, note that for any i ∈ N, , and that if ∈ Γ , then Γ stoL α (A) may not contain the trivial eigenvalue (also see Table 1). So, by these examples, we conclude that the set in Theorem 2.2 captures all eigenvalues di erent from of a stochastic matrix more precisely than the sets in Theorem 1.1 and Theorem 1.2 in some cases.
The set Γ [ , ] (A) in Remark 2.5 is not of much practical use because it involves some parameters α i . In fact, we can take some special α i in practice, which is illustrated by the following example. Table 1, we have that

Example 2.6. Consider the third stochastic matrix A in Example 2.3. By
which is shown in Figure 1, where Γ stoL α (A) is drawn slightly thicker than Γ . Furthermore, we take the rst vectors ], j = , , generated by the MATLAB code alpha = rand( , ), that is, , . ].

By Remark 2.5, we have that for any
We draw this set in the complex plane, see Figure 2. It is easy to see and This example shows that we can take some special α i to get a set which is tighter than the sets in Theorems 1.1 and 1.2. It is well-known that an eigenvalue inclusion set leads to a su cient condition for nonsingular matrices, and vice versa [12,16]. Hence, from Theorem 2.2 or Remark 2.5, we can get a nonsingular condition for stochastic matrices.
where CLᾱ i i is de ned as (2), then A is nonsingular.
Proof. Suppose that A is singular, that is, ∈ σ(A). From Theorem 2.2, we have that for any α i ∈ [ , ], i ∈ N, In particular, Hence, there is an index i ∈ N such that

An upper bound for the moduli of subdominant eigenvalues
By using the set Γ stoL α (A) in Theorem 2.2, we can give a bound to estimate the moduli of subdominant eigenvalues of a stochastic matrix. where Proof. Let Therefore, each f i (α i ), i ∈ N is a continuous function of α i ∈ [ , ], and there areα i ∈ [ , ], i ∈ N such that For theseα i ∈ [ , ], i ∈ N, by Theorem 2.2 we have Hence, there is an index i ∈ N such that By (5) we have λ ≤ min The conclusion follows.
As in the proof of Theorem 3.1, we can give another bound to estimate the moduli of subdominant eigenvalues by using the sets Γ stol (A), Γ stov (A) and Γ stoq (A)in Theorem 1.3, Γ stoL (A) and Γ stoV (A) in Theorem 1.4, respectively. where Proof. We rst prove λ ≤ ρ L . From Theorem 1.4, As in the proof of Theorem 3.1, we have that there is an index i ∈ N such that i.e., λ ≤ ρ L . Similarly, by we can get respectively λ ≤ ρ V , λ ≤ ρ q , λ ≤ ρ v , and λ ≤ ρ l .
The conclusion follows.
By the choices of α i in Remark 2.4, it is easy to get the relationships between ρ [ , ] , ρ L , ρ V , ρ q , ρ v and ρ l as follows.
As in the proof of Theorem 3.2, by Theorems 1.1 and 1.2 two upper bounds for the subdominant eigenvalue of a stochastic matrix are obtained easily.
For the comparison of ρ [ , ] and the upper bound (8), we conclude here that by taking some special α i and the fact that Λ is given by Theorems 1.1 and 1.2, an upper bound can be obtained, which is better than

Special choices of α i for the set Γ stoL α (A)
In this section, we choose α i for the set Γ stoL α (A) to give a set, which is tighter than the sets Γ stol (A) and where ρ , = max max Proof. Note that Hence, from Theorem 3.1, we have Furthermore, let .
at α = . Therefore, Inequality (10) is equivalent to The conclusion follows.
By the proof of Proposition 4.1, it is not di cult to see that the upper bound ρ , is larger than ρ [ , ] in Theorem 3.1, but ρ , depends only on the entries of a stochastic matrix. Moreover, ρ , ≤ ρ L and ρ , ≤ ρ l , which are given as follows. Proof. By the proof of Proposition 4.1, we have that ρ , is equivalent to the last of Inequality (10), that is, Since f (α) is increasing when ∆ i (A) ≥ , we have Then similarly as in the proof of ρ , ≤ ρ L , we can obtain easily ρ , ≤ ρ l . For the case that L i (A) = l i (A) for some i ∈ N, we have ∆ i (A) = and Similarly as in the case L i (A) > l i (A), i ∈ N, we can also obtain easily

Conclusions
In this paper, a set with n parameters in [ , ] is given to localize all eigenvalues di erent from for a stochastic matrix A, that is,