In this section, we present a new eigenvalue inclusion set by excluding some proper subsets that contain no eigenvalues of matrices from DZ sets, and, as an application, we provide a sufficient condition to judge the non-singularity of matrices.
Theorem 5
Let
\(A=[a_{ij}]\in \mathbb{C}^{n\times n}\). Then
$$\begin{aligned}& \sigma (A)\subseteq \varOmega (A)=\bigcap_{i\in N} \bigcup_{j\in N, j \neq i} \bigl(D_{ij}(A)\setminus \varOmega _{ij}(A) \bigr), \end{aligned}$$
where
$$\begin{aligned}& \varOmega _{ij}(A)=\bigl\{ z\in \mathbb{C}: \vert z-a_{ii} \vert \bigl( \vert z-a_{jj} \vert +r_{j}^{i}(A)\bigr)< \bigl(2 \vert a _{ij} \vert -r_{i}(A)\bigr) \vert a_{ji} \vert \bigr\} . \end{aligned}$$
Proof
Let λ be an eigenvalue of A and \(x=(x_{1},\ldots ,x_{n})^{T} \in \mathbb{C}^{n}\setminus \{0\}\) be its eigenvector. Then
$$\begin{aligned}& Ax=\lambda x. \end{aligned}$$
(1)
Let \(|x_{p}|=\max _{j\in N}|x_{j}|\). Obviously, \(|x_{p}|>0\). For any \(i\in N\), \(i\neq p\), the pth formula of (1) can be written as
$$\begin{aligned}& (\lambda -a_{pp})x_{p}=\sum _{s\neq p,i}a_{ps}x_{s}+a_{pi}x_{i}. \end{aligned}$$
(2)
Taking the absolute value of (2) and using the triangle inequality, we have
$$ \vert \lambda -a_{pp} \vert \vert x_{p} \vert \leq \sum_{s\neq p,i} \vert a_{ps} \vert \vert x_{s} \vert + \vert a_{pi} \vert \vert x _{i} \vert \leq r_{p}^{i}(A) \vert x_{p} \vert + \vert a_{pi} \vert \vert x_{i} \vert , $$
i.e.,
$$ \bigl( \vert \lambda -a_{pp} \vert -r_{p}^{i}(A)\bigr) \vert x_{p} \vert \leq \vert a_{pi} \vert \vert x_{i} \vert . $$
(3)
If \(|x_{i}|>0\), then from the ith formula of (1), i.e.,
$$\begin{aligned}& (\lambda -a_{ii})x_{i}=\sum _{t\neq i}a_{it}x_{t}, \end{aligned}$$
(4)
we have
$$\begin{aligned}& \vert \lambda -a_{ii} \vert \vert x_{i} \vert \leq r_{i}(A) \vert x_{p} \vert . \end{aligned}$$
(5)
Multiplying (3) and (5) and noting that \(|x_{p}||x_{i}|>0\), we have
$$\begin{aligned}& \vert \lambda -a_{ii} \vert \bigl( \vert \lambda -a_{pp} \vert -r_{p}^{i}(A)\bigr) \leq r_{i}(A) \vert a_{pi} \vert . \end{aligned}$$
(6)
If \(|x_{i}|=0\) in (3), then \(|\lambda -a_{pp}|-r _{p}^{i}(A)\leq 0\) as \(|x_{p}|>0\), which implies that (6) also holds. Therefore, \(\lambda \in D_{ip}(A)\). On the other hand, from (2) and (4), we have
$$\begin{aligned}& a_{pi}x_{i}=(\lambda -a_{pp})x_{p}- \sum_{s\neq p,i}a_{ps}x_{s} \end{aligned}$$
and
$$\begin{aligned}& a_{ip}x_{p}=(\lambda -a_{ii})x_{i}- \sum_{t\neq i,p}a_{it}x_{t}, \end{aligned}$$
which leads to
$$\begin{aligned} \vert a_{pi} \vert \vert x_{i} \vert \leq & \vert \lambda -a_{pp} \vert \vert x_{p} \vert +\sum_{s\neq p,i} \vert a _{ps} \vert \vert x_{s} \vert \leq \vert \lambda -a_{pp} \vert \vert x_{p} \vert +\sum _{s\neq p,i} \vert a_{ps} \vert \vert x _{p} \vert \\ =&\bigl( \vert \lambda -a_{pp} \vert +r_{p}^{i}(A)\bigr) \vert x_{p} \vert \end{aligned}$$
(7)
and
$$\begin{aligned}& \vert a_{ip} \vert \vert x_{p} \vert \leq \vert \lambda -a_{ii} \vert \vert x_{i} \vert + \sum_{t\neq i,p} \vert a_{it} \vert \vert x _{t} \vert \leq \vert \lambda -a_{ii} \vert \vert x_{i} \vert +r_{i}^{p}(A)) \vert x_{p} \vert , \end{aligned}$$
i.e.,
$$\begin{aligned}& \bigl( \vert a_{ip} \vert -r_{i}^{p}(A) \bigr) \vert x_{p} \vert \leq \vert \lambda -a_{ii} \vert \vert x_{i} \vert . \end{aligned}$$
(8)
By (7) and (8), we have
$$\begin{aligned}& \vert a_{pi} \vert \bigl(2 \vert a_{ip} \vert -r_{i}(A)\bigr)\leq \vert \lambda -a_{ii} \vert \bigl( \vert \lambda -a_{pp} \vert +r _{p}^{i}(A) \bigr), \end{aligned}$$
which implies that \(\lambda \notin \varOmega _{ip}(A)\). Hence, \(\lambda \in (D_{ip}(A)\setminus \varOmega _{ip}(A) )\).
For some certain \(i\in N\), \(i\neq p\), since we do not know which p is appropriate to λ, we can only conclude that
$$\begin{aligned}& \lambda \in \bigcup_{p\in N, i \neq p} \bigl(D_{ip}(A) \setminus \varOmega _{ip}(A) \bigr). \end{aligned}$$
Furthermore, by the arbitrariness of i, we have
$$\begin{aligned}& \lambda \in \bigcap_{i\in N}\bigcup _{j\in N, j\neq i} \bigl(D_{ij}(A) \setminus \varOmega _{ij}(A) \bigr). \end{aligned}$$
The conclusion follows. □
Now, a comparison theorem for Theorems 1, 2 and 5 is obtained.
Theorem 6
Let
\(A=[a_{ij}]\in \mathbb{C}^{n\times n}\). Then
$$\begin{aligned}& \varOmega (A)\subseteq D(A)\subseteq \varGamma (A). \end{aligned}$$
Proof
It is showed in Theorem 7 of [19] that \(D(A)\subseteq \varGamma (A)\). For any \(i,j\in N\), \(j\neq i\), by \((D_{ij}(A)\setminus \varOmega _{ij}(A)) \subseteq D_{ij}(A)\), obviously,
$$\begin{aligned}& \bigcap_{i\in N}\bigcup_{j\in N, i\neq j} \bigl(D_{ij}(A)\setminus \varOmega _{ij}(A)\bigr)\subseteq \bigcap_{i\in N}\bigcup_{j\in N, i\neq j}D_{ij}(A). \end{aligned}$$
Hence, \(\varOmega (A)\subseteq D(A)\) holds. □
Next, based on the fact that \(\det (A)=0\) if and only if \(0\in \sigma (A)\) for a matrix A, we can obtain the following condition for judging the non-singularity of matrices.
Corollary 1
Let
\(A=[a_{ij}]\in \mathbb{C}^{n\times n}\). If there exists
\(i\in N\), for any
\(j\in N\), \(j\neq i\), either
$$\begin{aligned}& \vert a_{ii} \vert \bigl( \vert a_{jj} \vert -r_{j}^{i}(A) \bigr)>r_{i}(A) \vert a_{ji} \vert , \end{aligned}$$
(9)
or
$$\begin{aligned}& \vert a_{ii} \vert \bigl( \vert a_{jj} \vert +r_{j}^{i}(A)\bigr)< \bigl(2 \vert a_{ij} \vert -r_{i}(A)\bigr) \vert a_{ji} \vert , \end{aligned}$$
(10)
then
A
is non-singular.
Proof
Let \(0\in \sigma (A)\). By Theorem 5, we have \(0\in \varOmega (A)\), i.e., for each \(i\in N\), there exists \(j\in N\), \(j \neq i\), such that
$$\begin{aligned}& \vert a_{ii} \vert \bigl( \vert a_{jj} \vert -r_{j}^{i}(A)\bigr)\leq r_{i}(A) \vert a_{ji} \vert \end{aligned}$$
and
$$\begin{aligned}& \vert a_{ii} \vert \bigl( \vert a_{jj} \vert +r_{j}^{i}(A)\bigr)\geq \bigl(2 \vert a_{ij} \vert -r_{i}(A)\bigr) \vert a_{ji} \vert . \end{aligned}$$
This contradicts (9) and (10). Consequently, \(0\notin \sigma (A)\), that is, A is non-singular. □
Remark 1
(i) Let i and j be any two elements of N, and \(i\neq j\). If \((2|a_{ij}|-r_{i}(A))|a_{ji}|>0\), then, by \(|a_{ij}|\leq r_{i}(A)\), we have \(0<2|a_{ij}|-r_{i}(A)\leq r_{i}(A)\) and
$$\begin{aligned} \vert z-a_{ii} \vert \bigl( \vert z-a_{jj} \vert -r^{i}_{j}(A)\bigr) \leq & \vert z-a_{ii} \vert \bigl( \vert z-a_{jj} \vert +r ^{i}_{j}(A) \bigr) \\ < &\bigl(2 \vert a_{ij} \vert -r_{i}(A)\bigr) \vert a_{ji} \vert \\ \leq & r_{i}(A) \vert a_{ji} \vert , \end{aligned}$$
which implies that
$$\begin{aligned}& \varOmega _{ij}(A)\subseteq D_{ij}(A) \quad \mbox{and}\quad \bigl(D_{ij}(A)\setminus \varOmega _{ij}(A) \bigr)\subseteq D_{ij}(A). \end{aligned}$$
(11)
That is to say, \(\varOmega _{ij}(A)\) is well defined. If \((2|a_{ij}|-r _{i}(A))|a_{ji}|\leq 0\), then \(\varOmega _{ij}(A)=\emptyset \). Obviously, (11) also holds. Here, \(\varOmega _{ij}(A)\) is called the exclusion set of \(D_{ij}(A)\).
(ii) As has been shown in [7, 8], the wider the class of non-singular matrices is, the tighter eigenvalue localization set it will lead to. Obviously, this conclusion, in turn, holds true. By Corollary 1, one can conclude that the conditions of Corollary 1 for judging the non-singularity of matrices are weaker than those in Theorems 3 and 4.
Next, an example is given to show that \(\varOmega (A)\) can catch all eigenvalues of a matrix A more precisely than \(\varGamma (A)\) and \(D(A)\), and that Theorems 3 and 4 cannot be used to judge the non-singularity of A in some cases, but Corollary 1 works better.
Example 1
Let
$$A=\begin{bmatrix} 11&4+\mathbf{{i}}&0&15-\mathbf{{i}} \\ -2-\mathbf{{i}}&10&5-\mathbf{{i}}&0 \\ 0&6&12+\mathbf{{i}}&4 \\ 16&2&0&11 \end{bmatrix}. $$
By computations, all eigenvalues of \(\mathcal{A}\) are \(26.4293-0.7552\mathbf{{i}}\), \(-4.4930+0.6450\mathbf{{i}}\), \(16.0853+0.0217\mathbf{{i}}\), \(5.9784+1.0885\mathbf{{i}}\). Next, the eigenvalue location and the determination of non-singularity for A are considered.
(I) Eigenvalue inclusion sets for A.
From Theorem 1, we have
$$\begin{aligned}& \varGamma (A)= \bigl\{ z\in \mathbb{C}: \vert z-11 \vert \leq 19.1564 \bigr\} . \end{aligned}$$
From Theorem 2, we have
$$\begin{aligned}& D(A)=D_{24}(A)= \bigl\{ z\in \mathbb{C}: \vert z-10 \vert \bigl( \vert z-11 \vert -16 \bigr)\leq 15.0662 \bigr\} . \end{aligned}$$
From Theorem 5, we have
$$\begin{aligned}& \varOmega (A)= \bigl[D_{12}(A)\cup D_{13}(A) \cup \bigl(D_{24}(A)\setminus \varOmega _{41}(A)\bigr) \cup \bigl(D_{43}(A)\cap \varOmega _{14}(A)\bigr)\bigr], \end{aligned}$$
where
$$\begin{aligned}& D_{12}(A)= \bigl\{ z\in \mathbb{C}: \vert z-11 \vert \bigl( \vert z-10 \vert -5.0990 \bigr)\leq 42.8356 \bigr\} , \\& D_{13}(A)= \bigl\{ z\in \mathbb{C}: \vert z-11 \vert \bigl( \vert z-12-\mathbf{{i}} \vert -10 \bigr)\leq 0 \bigr\} , \\& D_{43}(A)= \bigl\{ z\in \mathbb{C}: \vert z-11 \vert \bigl( \vert z-12-\mathbf{{i}} \vert -18 \bigr)\leq 72 \bigr\} , \\& \varOmega _{14}(A)= \bigl\{ z\in \mathbb{C}: \vert z-11 \vert \bigl( \vert z-11 \vert +2 \bigr)\leq 175.568 \bigr\} , \\& \varOmega _{41}(A)= \bigl\{ z\in \mathbb{C}: \vert z-11 \vert \bigl( \vert z-11 \vert +4.1231 \bigr)\leq 210.4662 \bigr\} . \end{aligned}$$
The eigenvalue localization sets \(\varGamma (A)\), \(D(A)\) and \(\varOmega (A)\) are drawn in Fig. 1, respectively, as red boundary, black boundary and yellow zones, and all eigenvalues are plotted as red asterisks. It is obvious that
$$\begin{aligned} \sigma (A)\subseteq \varOmega (A)\subseteq D(A)\subset \varGamma (A), \end{aligned}$$
another way of stating it is, \(\varOmega (A)\) can capture all eigenvalues of A more precisely than \(D(A)\) and \(\varGamma (A)\).
(II) The determination for non-singularity of A.
From Fig. 1, one can see that \(0\in \varGamma (A)\) and \(0\in D(A)\), but \(0\notin \varOmega (A)\), that is, the sets \(\varGamma (A)\) and \(D(A)\) cannot be used to judge the non-singularity of A. However, by \(0\notin \varOmega (A)\), one can conclude that A is non-singular.
Furthermore, as
$$\begin{aligned} \vert a_{11} \vert =11< 19.1564= r_{1}(A) \end{aligned}$$
and
$$\begin{aligned}& \vert a_{11} \vert \bigl( \vert a_{44} \vert -r_{4}^{1}(A) \bigr)=99.0000 < 306.5024=r_{1}(A) \vert a _{41} \vert , \\& \vert a_{22} \vert \bigl( \vert a_{44} \vert -r_{4}^{2}(A) \bigr)=-50.0000 < 14.6702=r_{2}(A) \vert a _{42} \vert , \\& \vert a_{33} \vert \bigl( \vert a_{44} \vert -r_{4}^{3}(A) \bigr)=-84.2912 < 0=r_{3}(A) \vert a _{43} \vert , \\& \vert a_{44} \vert \bigl( \vert a_{33} \vert -r_{3}^{4}(A) \bigr)= 66.4575 < 72.0000 =r _{4}(A) \vert a_{34} \vert , \end{aligned}$$
we know that the conditions in Theorems 3 and 4 do not hold, that is, Theorems 3 and 4 do not work. However, by
$$\begin{aligned}& \vert a_{11} \vert \bigl( \vert a_{22} \vert -r_{2}^{1}(A) \bigr)=53.9108 > 42.8350=r_{1}(A) \vert a _{21} \vert , \\& \vert a_{11} \vert \bigl( \vert a_{33} \vert -r_{3}^{1}(A) \bigr)=22.4575 > 0= r_{1}(A) \vert a _{31} \vert , \\& \vert a_{11} \vert \bigl( \vert a_{44} \vert +r_{4}^{1}(A) \bigr)=143.0000 < 174.5631=\bigl(2 \vert a _{14} \vert - r_{1}(A)\bigr) \vert a_{41} \vert , \end{aligned}$$
we know that there exists an index \(i=1\), for \(j=2,3,4\), either (9) or (10) holds. Then we can conclude the non-singularity of A by Corollary 1.