A new localization set for generalized eigenvalues

A new localization set for generalized eigenvalues is obtained. It is shown that the new set is tighter than that in (Numer. Linear Algebra Appl. 16:883-898, 2009). Numerical examples are given to verify the corresponding results.


Introduction
Let C n×n denote the set of all complex matrices of order n. For the matrices A, B ∈ C n×n , we call the family of matrices A -zB a matrix pencil, which is parameterized by the complex number z. Next, we regard a matrix pencil A -zB as a matrix pair ( We now list some notation which will be used in the following. Let N = {, , . . . , n}. Given two matrices A = (a ij ), B = (b ij ) ∈ C n×n , we denote  Recently, in [], Nakatsukasa presented a different Geršgorin-type theorem to estimate all generalized eigenvalues of a matrix pair (A, B) for the case that the ith row of either A (or B) is SDD for any i ∈ N . Although the set obtained by Nakatsukasa is simpler to compute than that in Theorem , the set is not tighter than that in Theorem  in general.
In this paper, we research the generalized eigenvalue localization for a regular matrix pair (A, B) without the restrictive assumption that the ith row of either A (or B) is SDD for any i ∈ N . By considering Ax = λBx and using the triangle inequality, we give a new inclusion set for generalized eigenvalues, and then prove that this set is tighter than that in Theorem  (Theorem  of []). Numerical examples are given to verify the corresponding results.

Main results
In this section, a set is provided to locate all the generalized eigenvalue of a matrix pair. Next we compare the set obtained with the generalized Geršgorin set in Theorem .

A new generalized eigenvalue localization set
Without loss of generality, let Solving for x p and x q in () and (), we obtain Taking absolute values of () and () and using the triangle inequality yield |a pkλb pk ||x k |.
Since x p =  and x q =  are, in absolute value, the largest and second largest components of x, respectively, we divide through by their absolute values to obtain and (a ppλb pp )(a qqλb qq ) -(a pqλb pq )(a qpλb qp ) ≤ |a ppλb pp |R p q (A, B, λ) + |a qpλb qp |R q p (A, B, λ).
(ii) If x q = , then x p is the only nonzero entry of x. From equality (), we have which implies that, for any i ∈ N , a ip = λb ip , i.e., a ipλb ip = . Hence for any i ∈ N , i = p, Moreover, when inequalities () and () hold, the matrix B is singular, and det(A -zB) has degree less than n. As we are considering regular matrix pairs, the degree of the polynomial det(A -zB) has to be at least one; thus, at least one of the sets ij (A, B) ∩ ji (A, B) has to be nonempty, implying that the set (A, B) of a regular matrix pair is always nonempty.
We now establish the following properties of the set (A, B).

Comparison with the generalized Geršgorin set
We now compare the set in Theorem  with the generalized Geršgorin set in Theorem . First, we observe two examples in which the generalized Geršgorin set is an unbounded set or the entire complex plane.

Example  Let
It is easy to see that b  = , β() = {} and Hence, from the part (iii) of Theorem , we see that We establish their comparison in the following. Next, we prove that then Then from inequalities () and (), we have If a ij = zb ij , then from z ∈ ij (A, B), we have (A, B, z).
Moreover, from inequality (), we obtain |a jjzb jj | = . It is obvious that If a ij = zb ij , then from inequality (), we have Hence, () holds.
(ii) Similar to the proof of (i), we also see that, for z ∈ ji (A, B), () holds. The conclusion follows from (i) and (ii).
Since the matrix pairs (A, B) and (A T , B T ) have the same generalized eigenvalues, we can obtain a theorem by applying Theorem  to (A T , B T ).
It is easy to see that B is SDD. Hence, from the part (ii) of Theorem , we see that