An Eigenvalue Inclusion Set for Matrices with a Constant Main Diagonal Entry

: A set to locate all eigenvalues for matrices with a constant main diagonal entry is given, and it is proved that this set is tighter than the well-known Geršgorin set, the Brauer set and the set proposed in (Linear and Multilinear Algebra, 60:189-199, 2012). Furthermore, by applying this result to Toeplitz matrices as a subclass of matrices with a constant main diagonal, we obtain a set including all eigenvalues of Toeplitz matrices.


Introduction
Eigenvalue localization is an important topic in Matrix theory and its applications. Many eigenvalue inclusion sets for a matrix A = [a ij ] ∈ C n×n [1][2][3][4][5][6][7][8][9][10][11] have been established, such as the well-known Geršgorin set [5,11] and the Brauer set [1,11]. However, as Melman [9] pointed out, for the special class of matrices with a constant main diagonal (c.m.d.), both the Geršgorin and Brauer sets each consists of a single disc, a rather uninteresting outcome. In fact, if a matrix A = [a ij ] ∈ C n×n satisfies a 11 = a 22 = · · · = a nn =ā, then both Γ(A) and K(A) reduce, respectively, to the following forms: Obviously, the Geršgorin and Brauer sets are just discs [9].
To localize all eigenvalues of matrices with a c.m.d. more precisely, Melman also [9] gave an eigenvalue inclusion set (see Theorem 1), which is tighter than Γ(A) and K(A). Theorem 1 ([9] Theorem 2.1). Let A = [a ij ] ∈ C n×n with a ii =ā for all i ∈ N, n ≥ 2. Let σ(A) be the spectrum of the matrix A, that is, σ(A) = {λ ∈ C : det(λI − A) = 0}. Then, and In [7], Li and Li provided two tighter sets including all eigenvalues of a matrix with a c.m.d. (see Theorems 2 and 3). where where Furthermore, In this paper, we first give a sufficient condition for non-singular matrices, which leads to a new set including all eigenvalues of matrices with a c.m.d. As an application, in Section 3, we apply the result obtained in Section 2 to Toeplitz matrices as a subclass of matrices with a c.m.d. and obtain a new eigenvalue inclusion set. All the new eigenvalue inclusion sets are proved to be tighter than those in [9].

A New Eigenvalue Inclusion Set for Matrices with a c.m.d.
In this section, we present a new eigenvalue inclusion set for matrices with a c.m.d. First, a sufficient condition for non-singular matrices is given.

Lemma 1.
For any A = [a ij ] ∈ C n×n with a ii =ā for all i ∈ N, and n ≥ 2, if where A 0 = A −āI, then A is non-singular.
From Lemma 1, we can obtain a new eigenvalue inclusion set for matrices with a c.m.d.
From Lemma 1, we have that λI − A is non-singular. This contradicts that λI − A is singular. Hence, λ ∈Ω(A).
We now give a comparison between the new eigenvalue setΩ(A) and the set Ω(A) in Theorem 1.

Theorem 5.
Let A = [a ij ] ∈ C n×n with a ii =ā for any i ∈ N, and n ≥ 2. Then, If (8), As Equation (9) holds for any i and j (i = j) in N, thereforeΩ(A) ⊆ Ω(A). This example shows that the the new eigenvalue inclusion set in Theorem 4 is tighter than the Geršgorin set Γ(A), the Brauer set K(A) and the set Ω(A) obtained in [9].  and

Example 1. Consider the matrix A (the matrix A 4 in [9]),
Note here that Ω 1 (A) = Ω 1 (A T ) and Ω 2 (A) = Ω 2 (A T ). However, the sets Ω 2 (A) and Ω(A) Ω (A T ) (also Ω 1 (A) andΩ(A) Ω (A T )) cannot be compared with each other. In fact, we also consider the matrix A in Example 1, and draw Ω 2 (A), andΩ(A) Ω (A T ) in Figures 2 and 3. It is not difficult to see that

Eigenvalue Inclusion Set for Toeplitz Matrices
Toeplitz matrices, a subclass of matrices with a c.m.d., arise in many fields of application [12][13][14][15][16][17][18], such as probability and statistics, signal processing, differential and integral equations, Markov chains, Padé approximation, etc. For example, consider an assigned Lebesgue integrable function f defined on the fundamental interval I = [−π, π) and periodically extended to the whole real axis, and the Fourier coefficients a k of f that is where k is an integer number. From the coefficients a k one can build the infinite dimensional Toeplitz matrix Tn( f ) with entries (Tn( f )) st = a s−t , s, t = 1, 2 . . . , n [12,13,16]. Toeplitz matrices are constant along all their NW-SE diagonals [7,9], i.e., a Toeplitz matrix T ∈ C n×n has the following form: Indeed, if f is a real valued function, we have a k =ā −k and, consequently, T n ( f ) is Hermitian; moreover, if f (x) = f (−x), then the coefficients a k are real and T n ( f ) is symmetric. The following result can be found in [12,19] and in a multilevel setting in [16,17]. Furthermore, there exist further results establishing precisely how fast the convergence holds [13,17]. Since in applications (differential and fractional operators/equations, shift-invariant integral operators/equations, signal and image processing etc.) often the underlying Toeplitz matrices have large size n, then the results in [12,13,16,17] are difficult to beat and improved. When f is complex-values the theory is more complicated and in that case the convex hull of the essential range of f plays a role (see [13,18]). Obviously, a Toeplitz matrix is persymmetric. Here, we call A persymmetric if A is symmetric with respect to the main anti-diagonal [9]. Furthermore, the square of a Toeplitz matrix T is not necessary Toeplitz, but it is persymmetric.
In [9], Melman applied the eigenvalue inclusion Theorem (Theorem 1) of matrices with a c.m.d. to Toeplitz matrices, and obtained the following simpler form of the eigenvalue inclusion set. Theorem 7 ([9] Theorem 3.1). Let T = [t ij ] ∈ C n×n be a Toeplitz matrix and t ii =t, n ≥ 2. Then, where 2 , i f n is odd.
Theorem 8. Let T = [t ij ] ∈ C n×n be a Toeplitz matrix with t 11 =t and n ≥ 2. Then,
From Theorems 5, 7 and 8, we can obtain easily the comparison results as follows.
Theorem 9. Let T = [t ij ] ∈ C n×n be a Toeplitz matrix with t 11 =t and n ≥ 2. Then,
This example shows that the new eigenvalue inclusion set in Theorem 8 is tighter than the set obtained in [9], the Geršgorin set and the Brauer set for a Toeplitz matrix.

Conclusions
In this paper, we obtain a new eigenvalue inclusion set for matrices with a c.m.d. We then apply this result to Toeplitz matrices, and get a set including all eigenvalues of Toeplitz matrices. Although they needs more computations to obtain the new eigenvalue sets than those in [9], the new sets capture all eigenvalues more precisely than those in [9].
Author Contributions: All the authors inferred the main conclusions and approved the current version of this manuscript.