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 denote the set of all complex matrices of order n. For the matrices , we call the family of matrices a matrix pencil, which is parameterized by the complex number z. Next, we regard a matrix pencil as a matrix pair [1]. A matrix pair is called regular if , and otherwise singular. A complex number λ is called a generalized eigenvalue of , if Furthermore, we call a nonzero vector a generalized eigenvector of associated with λ if Let denote the generalized spectrum of . Clearly, if B is an identity matrix, then reduces to the spectrum of A, i.e. . When B is nonsingular, is equivalent to the spectrum of , that is, So, in this case, has n generalized eigenvalues. Moreover, if B is singular, then the degree of the characteristic polynomial is , so the number of generalized eigenvalues of the matrix pair is d, and, by convention, the remaining eigenvalues are ∞ [1, 2].

We now list some notation which will be used in the following. Let . Given two matrices , , we denote and

The generalized eigenvalue problem arises in many scientific applications; see [3-5]. Many researchers are interested in the localization of all generalized eigenvalues of a matrix pair; see [1, 2, 6, 7]. In [1], Kostić et al. provided the following Geršgorin-type theorem of the generalized eigenvalue problem.

Theorem 1

[1], Theorem 7

Let , and be a regular matrix pair. Then

Here, is called the generalized Geršgorin set of a matrix pair and the ith generalized Geršgorin set. As showed in [1], is a compact set in the complex plane if and only if B is strictly diagonally dominant (SDD) [8]. When B is not SDD, may be an unbounded set or the entire complex plane (see Theorem 2).

Theorem 2

[1], Theorem 8

Let , , . Then the following statements hold:

Let be such that, for at least one , . Then is an unbounded set in the complex plane if and only if .

is a compact set in the complex plane if and only if B is SDD, that is, .

If there is an index such that both and where , then , and consequently , is the entire complex plane.

Recently, in [2], Nakatsukasa presented a different Geršgorin-type theorem to estimate all generalized eigenvalues of a matrix pair for the case that the ith row of either A (or B) is SDD for any . Although the set obtained by Nakatsukasa is simpler to compute than that in Theorem 1, the set is not tighter than that in Theorem 1 in general.

In this paper, we research the generalized eigenvalue localization for a regular matrix pair without the restrictive assumption that the ith row of either A (or B) is SDD for any . By considering 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 1 (Theorem 7 of [1]). 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 1.

A new generalized eigenvalue localization set

Theorem 3

Let , , with and be a regular matrix pair. Then

Proof

For any , let be an associated generalized eigenvector, i.e., Without loss of generality, let Then .

(i) If , then from Equality (1), we have and equivalently, and Solving for and in (2) and (3), we obtain and Taking absolute values of (4) and (5) and using the triangle inequality yield and Since and are, in absolute value, the largest and second largest components of x, respectively, we divide through by their absolute values to obtain and Hence,

(ii) If , then is the only nonzero entry of x. From equality (1), we have which implies that, for any , , i.e., . Hence for any , , From (i) and (ii), . The proof is completed. □

Since the matrix pairs and have the same generalized eigenvalues, we can obtain a theorem by applying Theorem 3 to .

Theorem 4

Let , , with , and be a regular matrix pair. Then

Remark 1

If B is an identity matrix, then Theorems 3 and 4 reduce to the corresponding results of [9].

Remark 2

When all entries of the ith and jth rows of the matrix B are zero, then and Hence, if and then otherwise, Moreover, when inequalities (6) and (7) hold, the matrix B is singular, and has degree less than n. As we are considering regular matrix pairs, the degree of the polynomial has to be at least one; thus, at least one of the sets has to be nonempty, implying that the set of a regular matrix pair is always nonempty.

We now establish the following properties of the set .

Theorem 5

Let , , with and be a regular matrix pair. Then the set contains zero if and only if inequalities (6) and (7) hold.

The conclusion follows directly from putting in the inequalities of and . □

Theorem 6

Let , , with and be a regular matrix pair. If there exist , , such that and where , then , and consequently is the entire complex plane.

The conclusion follows directly from the definitions of and . □

Comparison with the generalized Geršgorin set

We now compare the set in Theorem 3 with the generalized Geršgorin set in Theorem 1. First, we observe two examples in which the generalized Geršgorin set is an unbounded set or the entire complex plane.

Example 1

Let It is easy to see that and Hence, from the part (i) of Theorem 2, we see that is unbounded. However, the set in Theorem 3 is compact. These sets are given by Figure 1, where the actual generalized eigenvalues are plotted with asterisks. Clearly, .

Figure 1

of Example on the left, and on the right.

Example 2

Let It is easy to see that , and Hence, from the part (iii) of Theorem 2, we see that is the entire complex plane, but the set in Theorem 3 is not. is given by Figure 2, where the actual generalized eigenvalues are plotted with asterisks.

Figure 2

of Example .

We establish their comparison in the following.

Theorem 7

Let , , with and be a regular matrix pair. Then

Let . Then there are , such that Next, we prove that and

(i) For , then or . If , then (8) holds. If , that is, then Note that and . Then from inequalities (10) and (11), we have which implies that If , then from , we have Moreover, from inequality (10), we obtain . It is obvious that If , then from inequality (12), we have that is, Hence, (8) holds.

(ii) Similar to the proof of (i), we also see that, for , (9) holds.

The conclusion follows from (i) and (ii). □

Since the matrix pairs and have the same generalized eigenvalues, we can obtain a theorem by applying Theorem 7 to .

Theorem 8

Let , , with and be a regular matrix pair. Then

Example 3

[1], Example 1

Let It is easy to see that B is SDD. Hence, from the part (ii) of Theorem 2, we see that is compact. and are given by Figure 3, where the exact generalized eigenvalues are plotted with asterisks. Clearly, .

Figure 3

of Example on the left, and on the right.

Remark 3

From Examples 1, 2 and 3, we see that the set in Theorem 3 is tighter than that in Theorem 1 (Theorem 7 of [1]). In addition, note that A and B in Example 1 satisfy and respectively. Hence, we cannot use the method in [2] to estimate the generalized eigenvalues of the matrix pair (A,B). However, the set we obtain is very compact.

Conclusions

In this paper, we present a new generalized eigenvalue localization set , and we establish the comparison of the sets and in Theorem 7 of [1], that is, captures all generalized eigenvalues more precisely than , which is shown by three numerical examples.