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A new localization set for generalized eigenvalues
Journal of Inequalities and Applications volume 2017, Article number: 113 (2017)
Abstract
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.
1 Introduction
Let \(\mathbb{C}^{n\times n}\) denote the set of all complex matrices of order n. For the matrices \(A, B \in\mathbb{C}^{n\times n}\), 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 \((A, B)\) [1]. A matrix pair \((A, B)\) is called regular if \(\operatorname{det}(A-zB) \neq0 \), and otherwise singular. A complex number λ is called a generalized eigenvalue of \((A, B)\), if
Furthermore, we call a nonzero vector \(x\in\mathbb{C}^{n}\) a generalized eigenvector of \((A, B)\) associated with λ if
Let \(\sigma(A,B)=\{\lambda\in\mathbb{C}: \operatorname{det}(A-\lambda B)=0\}\) denote the generalized spectrum of \((A, B)\). Clearly, if B is an identity matrix, then \(\sigma(A,B)\) reduces to the spectrum of A, i.e. \(\sigma(A,B)=\sigma(A)\). When B is nonsingular, \(\sigma(A,B)\) is equivalent to the spectrum of \(B^{-1}A\), that is,
So, in this case, \((A, B)\) has n generalized eigenvalues. Moreover, if B is singular, then the degree of the characteristic polynomial \(\operatorname{det}(A-\lambda B)\) is \(d < n\), so the number of generalized eigenvalues of the matrix pair \((A, B)\) is d, and, by convention, the remaining \(n-d\) eigenvalues are ∞ [1, 2].
We now list some notation which will be used in the following. Let \(N=\{1,2,\ldots,n\}\). Given two matrices \(A=(a_{ij})\), \(B=(b_{ij})\in\mathbb{C}^{n\times n}\), 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 \(A,B\in\mathbb{C}^{n\times n}\), \(n\geq2\) and \((A,B)\) be a regular matrix pair. Then
Here, \(\Gamma(A,B)\) is called the generalized Geršgorin set of a matrix pair \((A, B)\) and \(\Gamma_{i}(A,B)\) the ith generalized Geršgorin set. As showed in [1], \(\Gamma(A,B)\) is a compact set in the complex plane if and only if B is strictly diagonally dominant (SDD) [8]. When B is not SDD, \(\Gamma(A,B)\) may be an unbounded set or the entire complex plane (see Theorem 2).
Theorem 2
[1], Theorem 8
Let \(A=(a_{ij})\), \(B=(b_{ij})\in\mathbb{C}^{n\times n}\), \(n\geq2\). Then the following statements hold:
-
(i)
Let \(i\in N\) be such that, for at least one \(j\in N\), \(b_{ij} \neq0\). Then \(\Gamma_{i} (A, B)\) is an unbounded set in the complex plane if and only if \(\vert b_{ii} \vert \leq r_{i}(B)\).
-
(ii)
\(\Gamma(A,B)\) is a compact set in the complex plane if and only if B is SDD, that is, \(\vert b_{ii} \vert > r_{i}(B)\).
-
(iii)
If there is an index \(i\in N\) such that both \(b_{ii} =0\) and
$$\vert a_{ii} \vert \leq\sum_{k\in\beta(i), \atop k\neq i} \vert a_{ik} \vert , $$where \(\beta(i)=\{j\in N: b_{ij}=0\}\), then \(\Gamma_{i}(A, B)\), and consequently \(\Gamma(A, B)\), is the entire complex plane.
Recently, in [2], 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 \(i\in N\). 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 \((A,B)\) without the restrictive assumption that the ith row of either A (or B) is SDD for any \(i\in N\). By considering \(Ax=\lambda Bx\) 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.
2 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.
2.1 A new generalized eigenvalue localization set
Theorem 3
Let \(A=(a_{ij})\), \(B=(b_{ij})\in\mathbb{C}^{n\times n}\), with \(n\geq2\) and \((A,B)\) be a regular matrix pair. Then
Proof
For any \(\lambda\in\sigma(A,B)\), let \(0\neq x=(x_{1},x_{2},\ldots,x_{n})^{T}\in \mathbb{C}^{n}\) be an associated generalized eigenvector, i.e.,
Without loss of generality, let
Then \(x_{p}\neq0\).
(i) If \(x_{q}\neq0\), then from Equality (1), we have
and
equivalently,
and
Solving for \(x_{p}\) and \(x_{q}\) in (2) and (3), we obtain
and
Taking absolute values of (4) and (5) and using the triangle inequality yield
and
Since \(x_{p}\neq0\) and \(x_{q}\neq0\) 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 \(x_{q}= 0\), then \(x_{p}\) is the only nonzero entry of x. From equality (1), we have
which implies that, for any \(i\in N\), \(a_{ip}=\lambda b_{ip}\), i.e., \(a_{ip}-\lambda b_{ip}=0\). Hence for any \(i\in N\), \(i\neq p\),
From (i) and (ii), \(\sigma(A,B)\subseteq\Phi(A,B)\). The proof is completed. □
Since the matrix pairs \((A,B)\) and \((A^{T},B^{T})\) have the same generalized eigenvalues, we can obtain a theorem by applying Theorem 3 to \((A^{T},B^{T})\).
Theorem 4
Let \(A=(a_{ij})\in\mathbb{C}^{n\times n}\), \(B=(b_{ij})\in \mathbb{C}^{n\times n}\), with \(n\geq2\), and \((A^{T},B^{T})\) 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 \(\operatorname{det}(A-zB)\) has degree less than n. As we are considering regular matrix pairs, the degree of the polynomial \(\operatorname{det}(A-zB)\) has to be at least one; thus, at least one of the sets \(\Phi_{ij}(A,B)\cap\Phi_{ji}(A,B)\) has to be nonempty, implying that the set \(\Phi(A,B)\) of a regular matrix pair is always nonempty.
We now establish the following properties of the set \(\Phi(A,B)\).
Theorem 5
Let \(A=(a_{ij})\), \(B=(b_{ij})\in\mathbb{C}^{n\times n}\), with \(n\geq2\) and \((A,B)\) be a regular matrix pair. Then the set \(\Phi_{ij}(A,B)\cap\Phi_{ji}(A,B)\) contains zero if and only if inequalities (6) and (7) hold.
Proof
The conclusion follows directly from putting \(z=0\) in the inequalities of \(\Phi_{ij}(A,B)\) and \(\Phi_{ji}(A,B)\). □
Theorem 6
Let \(A=(a_{ij})\), \(B=(b_{ij})\in\mathbb{C}^{n\times n}\), with \(n\geq2\) and \((A,B)\) be a regular matrix pair. If there exist \(i,j \in N\), \(i\neq j\), such that
and
where \(\beta(i)=\{k\in N: b_{ik}=0\}\), then \(\Phi_{ij}(A,B)\cap\Phi_{ji}(A,B)\), and consequently \(\Phi(A, B)\) is the entire complex plane.
Proof
The conclusion follows directly from the definitions of \(\Phi _{ij}(A,B)\) and \(\Phi_{ji}(A,B)\). □
2.2 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 \(b_{12}=0.1>0\) and
Hence, from the part (i) of Theorem 2, we see that \(\Gamma (A,B)\) is unbounded. However, the set \(\Phi(A,B)\) in Theorem 3 is compact. These sets are given by Figure 1, where the actual generalized eigenvalues are plotted with asterisks. Clearly, \(\Phi(A,B)\subset\Gamma(A,B)\).
Example 2
Let
It is easy to see that \(b_{11}=0\), \(\beta(1)=\{2\}\) and
Hence, from the part (iii) of Theorem 2, we see that \(\Gamma (A,B)\) is the entire complex plane, but the set \(\Phi(A,B)\) in Theorem 3 is not. \(\Phi(A,B)\) is given by Figure 2, where the actual generalized eigenvalues are plotted with asterisks.
We establish their comparison in the following.
Theorem 7
Let \(A=(a_{ij})\in\mathbb{C}^{n\times n}\), \(B=(b_{ij})\in\mathbb{C}^{n\times n}\), with \(n\geq2\) and \((A,B)\) be a regular matrix pair. Then
Proof
Let \(z\in\Phi(A,B)\). Then there are \(i,j \in N\), \(i \neq j\) such that
Next, we prove that
and
(i) For \(z\in\Phi_{ij}(A,B)\), then \(z\in\Gamma_{i}(A,B)\) or \(z\notin \Gamma_{i}(A,B)\). If \(z\in\Gamma_{i}(A,B)\), then (8) holds. If \(z\notin\Gamma_{i}(A,B)\), that is,
then
Note that \(R_{i}^{j}(A,B,z)=R_{i}(A,B,z)- \vert a_{ij}-z b_{ij} \vert \) and \(R_{j}^{i}(A,B,z)=R_{j}(A,B,z)- \vert a_{ji}-z b_{ji} \vert \). Then from inequalities (10) and (11), we have
which implies that
If \(a_{ij}=z b_{ij}\), then from \(z\in\Phi_{ij}(A,B)\), we have
Moreover, from inequality (10), we obtain \(\vert a_{jj}-z b_{jj} \vert =0\). It is obvious that
If \(a_{ij}\neq z b_{ij}\), then from inequality (12), we have
that is,
Hence, (8) holds.
(ii) Similar to the proof of (i), we also see that, for \(z\in \Phi_{ji}(A,B)\), (9) 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 7 to \((A^{T},B^{T})\).
Theorem 8
Let \(A=(a_{ij})\in\mathbb{C}^{n\times n}\), \(B=(b_{ij})\in\mathbb{C}^{n\times n}\), with \(n\geq2\) and \((A^{T},B^{T})\) 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 \(\Gamma(A,B)\) is compact. \(\Gamma(A,B)\) and \(\Phi(A,B)\) are given by Figure 3, where the exact generalized eigenvalues are plotted with asterisks. Clearly, \(\Phi(A,B)\subset\Gamma(A,B)\).
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.
3 Conclusions
In this paper, we present a new generalized eigenvalue localization set \(\Phi(A,B)\), and we establish the comparison of the sets \(\Phi(A,B)\) and \(\Gamma(A,B)\) in Theorem 7 of [1], that is, \(\Phi(A,B)\) captures all generalized eigenvalues more precisely than \(\Gamma(A,B)\), which is shown by three numerical examples.
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Acknowledgements
This work was supported by National Natural Science Foundations of China (Grant No.[11601473]) and CAS ‘Light of West China’ Program.
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Gao, J., Li, C. A new localization set for generalized eigenvalues. J Inequal Appl 2017, 113 (2017). https://doi.org/10.1186/s13660-017-1388-x
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DOI: https://doi.org/10.1186/s13660-017-1388-x