Abstract
In this paper, for a given matrix \(A=(a_{ij}) \in\mathbb{C}^{n\times n}\), in terms of \(r_{i}\) and \(c_{i}\), where \(r_{i} = \sum _{j = 1,j \ne i}^{n} {\vert {a_{ij} } \vert }\), \(c_{i} = \sum _{j = 1,j \ne i}^{n} {\vert {a_{ji} } \vert }\), some new inclusion sets for singular values of the matrix are established. It is proved that the new inclusion sets are tighter than the Geršgorin-type sets (Qi in Linear Algebra Appl. 56:105-119, 1984) and the Brauer-type sets (Li in Comput. Math. Appl. 37:9-15, 1999). A numerical experiment shows the efficiency of our new results.
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1 Introduction
Singular values and the singular value decomposition play an important role in numerical analysis and many other applied fields [3–8]. First, we will use the following notations and definitions. Let \(N :=\{1, 2, \ldots, n\}\), and assume \(n \geq2\) throughout. For a given matrix \(A=(a_{ij}) \in\mathbb{C}^{n\times n}\), we define \(a_{i} = |a_{ii } |\), \(s_{i} = \max\{r_{i}, c_{i} \} \) for any \(i \in N\) and \(u_{+} = \max\{0, u\}\), u is a real number, and where
In terms of \(s_{i}\), the Geršgorin-type, Brauer-type and Ky Fan-type inclusion sets of the matrix singular values are given in [1, 2, 9, 10], we list the results as follows.
Theorem 1
If a matrix \(A=(a_{ij}) \in\mathbb{C}^{n\times n}\), then
-
(i)
(Geršgorin-type, see [1]) all singular values of A are contained in
$$ C(A):=\bigcup _{i = 1}^{n} C_{i} \quad\textit{with } C_{i}=\bigl[(a_{i}-s_{i})_{+},(a_{i}+s_{i}) \bigr]\in R; $$(1) -
(ii)
(Brauer-type, see [2]) all singular values of A are contained in
$$ D(A):=\bigcup _{i = 1}^{n} \bigcup _{j = 1,j\neq i}^{n} \bigl\{ z\geq0: |z-a_{i}||z-a_{j}| \leq s_{i}s_{j}\bigr\} ; $$(2) -
(iii)
(Ky Fan-type, see [2]) let \(B=(b_{ij}) \in\mathbb{R}^{n\times n}\) be a nonnegative matrix satisfying \(b_{ij} \geq\max\{|a_{ij}|, |a_{ji}|\}\) for any \(i \neq j\), then all singular values of A are contained in
$$E(A):=\bigcup _{i = 1}^{n} \bigl\{ z\geq0: |z-a_{i}|\leq\rho(B)-b_{ii}\bigr\} . $$
We observe that all the results in Theorem 1 are based on the values of \(s_{i} = \max\{r_{i}, c_{i} \}\), if \(r_{i}\ll c_{i}\) or \(r_{i}\gg c_{i}\), all these singular value localization sets in Theorem 1 become very crude. In this paper, we give some new singular value localization sets which are based on the values of \(r_{i}\) and \(c_{i}\). The remainder of the paper is organized as follows. In Section 2, we give our main results. In Section 3, a numerical experiment is given to show the efficiency of our new results.
2 New inclusion sets for singular values
Based on the idea of Li in [2], we give our main results as follows.
Theorem 2
If a matrix \(A=(a_{ij}) \in\mathbb{C}^{n\times n}\), then all singular values of A are contained in
where
and
Proof
Let σ be an arbitrary singular value of A. Then there exist two nonzero vectors \(x = (x_{1}, x_{2}, \ldots, x_{n})^{T}\) and \(y = (y_{1}, y_{2},\ldots, y_{n})^{T}\) such that
Denote
Now, we assume that \(|x_{p}| \leq|y_{q}|\), the qth equations in (3) imply
Solving for \(y_{q}\) we can get
Taking the absolute value on both sides of the equation and using the triangle inequality yield
Then we can get
Similarly, if \(|y_{q}|\leq|x_{p}|\), we can get
Thus, we complete the proof. □
Remark 1
Since
and
the results in Theorem 2 are always better than the results in Theorem 1(i).
Theorem 3
If a matrix \(A=(a_{ij}) \in\mathbb{C}^{n\times n}\), then all singular values of A are contained in
where
and
Proof
Let σ be an arbitrary singular value of A. Then there exist two nonzero vectors \(x = (x_{1}, x_{2}, \ldots, x_{n})^{T}\) and \(y = (y_{1}, y_{2},\ldots, y_{n})^{T}\) such that
Denote \(\omega_{i} = \max\{|x_{i}|, |y_{i}|\}\). Let q be an index such that \(\omega_{q}= \max\{|\omega_{i}|, i \in N\}\). Obviously, \(\omega_{q}\neq0\). Let p be an index such that \(\omega_{p} = \max\{|\omega_{i}|, i \in N, i\neq q\}\).
Case I: We suppose \(\omega_{q}=|x_{q}|\), \(\omega_{p}=|x_{p}|\), similar to the proof of Theorem 2, the qth equations in (8) imply
Similarly, the pth equations in (8) imply
Multiplying inequalities (9) with (10), we have
Case II: We suppose \(\omega_{q}=|y_{q}|\), \(\omega_{p}=|y_{p}|\), similar to the proof of Theorem 2, the qth equations in (8) imply
Similarly, the pth equations in (8) imply
Multiplying inequalities (11) with (12), we have
Case III: We suppose \(\omega_{q}=|y_{q}|\), \(\omega_{p}=|x_{p}|\), similar to the proof of Theorem 2, the qth equations in (8) imply
Similarly, the pth equations in (8) imply
Multiplying inequalities (13) with (14), we have
Case IV: We suppose \(\omega_{q}=|x_{q}|\), \(\omega_{p}=|y_{p}|\), similar to the proof of Cases I, II, III, we can get
Thus, we complete the proof. □
Remark 2
Since
and
the results in Theorem 3 are always better than the results in Theorem 1(ii).
We now establish comparison results between \(\Gamma(A)\) and \(\Omega(A)\).
Theorem 4
If a matrix \(A=(a_{ij}) \in\mathbb{C}^{n\times n}\), then
Proof
Let z be any point of \(\Omega_{3} (A)\). Then there are \(i , j \in N\), \(i \neq j\), such that \(z\in\Omega_{3} (A)\), i.e.,
If \((\vert {a_{ii} } \vert r_{i} (A) + z c_{i} (A) ) ( \vert {a_{jj} } \vert c_{j} (A) + z r_{j} (A) )=0\), then
or
Therefore, \(z \in\Gamma_{1} (A)\cup\Gamma_{2} (A)\). Moreover, if \((\vert {a_{ii} } \vert r_{i} (A) + z c_{i} (A) ) ( \vert {a_{jj} } \vert c_{j} (A) + z r_{j} (A) )> 0\), then from inequality (15), we have
Hence, from inequality (16), we have that
or
That is, \(z \in\Gamma_{1} (A)\) or \(z \in\Gamma_{2} (A)\), i.e., \(z \in \Gamma(A)\).
Similarly, if z is any point of \(\Omega_{1} (A)\) or \(\Omega_{2} (A)\), we can get
and
Thus, we complete the proof. □
3 Numerical example
Example 1
Let
The singular values of A are \(\sigma_{1} = 4.1544\) and \(\sigma_{2} = 0.0241\). From Figure 1, it is easy to see that Theorem 2 is better than Theorem 1 for certain examples. In Figure 2, we can see that the results in Theorem 3 are tighter than the results in Theorem 2, which is analyzed in Theorem 4.
4 Conclusion
In this paper, some new inclusion sets for singular values are given. Theoretical analysis and numerical example show that these estimates are more efficient than recent corresponding results in some cases.
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Acknowledgements
He is supported by the Science and Technology Foundation of Guizhou province (Qian ke he Ji Chu [2016]1161); Guizhou Province Natural Science Foundation in China (Qian Jiao He KY [2016]255); the Doctoral Scientific Research Foundation of Zunyi Normal College (BS[2015]09). Liu is supported by the National Natural Science Foundation of China (71461027); Science and Technology Talent Training Object of Guizhou Province Outstanding Youth (Qian ke he ren zi [2015]06); Guizhou Province Natural Science Foundation in China (Qian Jiao He KY [2014]295); 2013, 2014 and 2015 Zunyi 15851 Talents Elite Project Funding; Zhunyi Innovative Talent Team (Zunyi KH(2015)38). Tian is supported by Guizhou Province Natural Science Foundation in China (Qian Jiao He KY [2015]451); the Science and Technology Foundation of Guizhou province (Qian ke he J zi [2015]2147). Ren is supported by the Science and Technology Foundation of Guizhou province (Qian ke he LH zi [2015]7006).
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He, J., Liu, YM., Tian, JK. et al. New inclusion sets for singular values. J Inequal Appl 2017, 64 (2017). https://doi.org/10.1186/s13660-017-1337-8
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DOI: https://doi.org/10.1186/s13660-017-1337-8