1 Introduction
Let \(M_{n}\) denote the set of all \(n\times n\) complex matrices and \(A=(a_{ij}), B=(b_{ij}) \in M_{n}\). If \(a_{ij}-b_{ij}\geq0\), we say that \(A\geq B\), and if \(a_{ij} \geq0\), we say that A is nonnegative, denoted by \(A\geq0\). The symbol \(\rho(A)\) stands for the spectral radius of A. If A is a nonnegative matrix, the Perron-Frobenius theorem guarantees that \(\rho(A)\in\sigma(A)\), where \(\sigma(A)\) denotes the spectrum of A.
If there does not exist a permutation matrix
P such that
$$P^{T}AP=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} A_{1}&A_{12}\\ 0&A_{2} \end{array}\displaystyle \right ), $$
where
\(A_{1}\),
\(A_{2}\) are square matrices, then
A is called irreducible.
Let A be an irreducible nonnegative matrix. It is well known that there exists a positive vector u such that \(Au=\rho(A)u\), u being called a right Perron eigenvector of A.
The Hadamard product of A, B is defined as \(A\circ B=(a_{ij}b_{ij})\in M_{n}\).
Let
\(A\geq0\),
\(B\geq0\). By using the Gersgorin theorem, Brauer theorem and Brualdi theorem, respectively, the authors of [
1‐
5] have given some inequalities for the upper bounds of
\(\rho(A\circ B)\). Audenaert [
6], Horn and Zhang [
7] proved a beautiful inequality on
\(\rho(A\circ B)\) for nonnegative matrices
A and
B, that is,
\(\rho(A\circ B)\leq\rho(A B)\). Huang [
8] generalized the above inequality to any
k nonnegative matrices, that is,
\(\rho(A_{1}\circ A_{2}\circ\cdots\circ A_{k})\leq\rho(A_{1}A_{2}\cdots A_{k})\). Motivated by [
8] and [
1‐
4,
9,
10], in this paper we propose some inequalities on the upper bounds for the spectral radius of the Hadamard product of any
k nonnegative matrices. These bounds generalize some existing results, and some comparisons between these bounds are also considered.
2 Main results
First, we give some lemmas which are useful for obtaining the main results.
Setting
\(k=2\) in Theorem
2.1, we have the following corollary.
Setting
\(k=2\) in Theorem
2.2, we have the following corollary.
We next give a simple comparison between the upper bound in (
2.1) and the upper bound in (
2.2). Without loss of generality, for
\(i\neq j\), assume that
$$\begin{aligned}& a_{ii}b_{ii}\cdots k_{ii}+\bigl( \rho(A_{1})-a_{ii}\bigr) \bigl(\rho(A_{2})-b_{ii} \bigr)\cdots \bigl(\rho(A_{k})-k_{ii}\bigr) \\& \quad \geq a_{jj}b_{jj} \cdots k_{jj}+\bigl( \rho(A_{1})-a_{jj}\bigr) \bigl(\rho (A_{2})-b_{jj} \bigr)\cdots\bigl(\rho(A_{k})-k_{jj}\bigr). \end{aligned}$$
Let
\(\gamma=a_{ii}b_{ii}\cdots k_{ii}+a_{jj}b_{jj}\cdots k_{jj}\). From (
2.2), we have
$$\begin{aligned}& a_{ii}b_{ii}\cdots k_{ii}+a_{jj}b_{jj} \cdots k_{jj}+ \bigl[(a_{ii}b_{ii}\cdots k_{ii}-a_{jj}b_{jj}\cdots k_{jj})^{2} \\& \qquad {}+ 4\bigl(\rho(A_{1})-a_{ii}\bigr) \bigl( \rho(A_{2})-b_{ii}\bigr)\cdots\bigl(\rho(A_{k})-k_{ii} \bigr) \bigl(\rho (A_{1})-a_{jj}\bigr) \\& \qquad {}\times{ \bigl(\rho(A_{2})-b_{jj}\bigr)\cdots\bigl( \rho(A_{k})-k_{jj}\bigr)} \bigr]^{\frac {1}{2}} \\& \quad \leq\gamma+ \bigl\{ (a_{ii}b_{ii}\cdots k_{ii}-a_{jj}b_{jj}\cdots k_{jj})^{2}+ 4\bigl(\rho(A_{1})-a_{ii}\bigr)\cdots\bigl( \rho(A_{k})-k_{ii}\bigr) \\& \qquad {}\times\bigl[\bigl( \rho(A_{1})-a_{ii}\bigr) \bigl( \rho(A_{2})-b_{ii}\bigr)\cdots\bigl(\rho (A_{k})-k_{ii} \bigr)+a_{ii}b_{ii}\cdots k_{ii}-a_{jj}b_{jj} \cdots k_{jj} \bigr]\bigr\} ^{\frac{1}{2}} \\& \quad =\gamma+\bigl[ \bigl(a_{ii}b_{ii}\cdots k_{ii}-a_{jj}b_{jj}\cdots k_{jj} +2\bigl( \rho(A_{1})-a_{ii}\bigr)\cdots\bigl(\rho(A_{k})-k_{ii} \bigr)\bigr)^{2} \bigr]^{\frac{1}{2}} \\& \quad =2a_{ii}b_{ii}\cdots k_{ii}+2\bigl( \rho(A_{1})-a_{ii}\bigr) \bigl(\rho (A_{2})-b_{ii} \bigr)\cdots\bigl(\rho(A_{k})-k_{ii}\bigr). \end{aligned}$$
Thus, we have
$$\begin{aligned}& \rho(A_{1}\circ A_{2}\circ\cdots\circ A_{k}) \\& \quad \leq\max_{ i\neq j} \frac{1}{2}\bigl\{ a_{ii} \cdots k_{ii}+a_{jj}\cdots k_{jj}+ \bigl[(a_{ii}\cdots k_{ii}-a_{jj}\cdots k_{jj})^{2} +4\bigl(\rho(A_{1})-a_{ii} \bigr) \\& \qquad {}\times\bigl(\rho(A_{2})-b_{ii}\bigr)\cdots\bigl( \rho(A_{k})-k_{ii}\bigr) \bigl(\rho (A_{1})-a_{jj} \bigr) \bigl(\rho(A_{2})-b_{jj}\bigr)\cdots\bigl( \rho(A_{k})-k_{jj}\bigr)\bigr]^{\frac {1}{2}}\bigr\} \\& \quad \leq\max_{ 1 \leq i \leq n} \frac{1}{2} \bigl[2a_{ii}b_{ii} \cdots k_{ii}+2\bigl(\rho(A_{1})-a_{ii}\bigr)\cdots \bigl(\rho(A_{k})-k_{ii}\bigr) \bigr] \\& \quad =\max_{ 1 \leq i \leq n} \bigl[a_{ii}b_{ii}\cdots k_{ii}+\bigl(\rho (A_{1})-a_{ii}\bigr) \bigl( \rho(A_{2})-b_{ii}\bigr)\cdots\bigl(\rho(A_{k})-k_{ii} \bigr) \bigr]. \end{aligned}$$
Hence, bound (
2.2) is better than bound (
2.1).
In [
8], the author proved that
$$ \rho(A_{1}\circ A_{2}\circ\cdots\circ A_{k})\leq\rho(A_{1}A_{2}\cdots A_{k}). $$
(2.3)
At present, we cannot give the comparison between bounds (
2.1) and (
2.3) or bounds (
2.2) and (
2.3), but the following numerical example shows that bounds (
2.1) and (
2.2) are better than (
2.3). Next,we give an example: Consider four
\(4\times4\) nonnegative matrices
$$\begin{aligned}& A=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 4&1&0&2\\ 0& 0.05&1&1\\ 0& 0&4 &0.5\\ 1 & 0.5&0&4 \end{array}\displaystyle \right ),\qquad B=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 1&1&1&1\\ 1&1&1&1\\ 1&1&1&1\\ 1&1&1&1 \end{array}\displaystyle \right ), \\& C=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 2 &0&1&1\\ 1&4&0.5&0.5\\ 1&0&3&0.5\\ 0.5&1&1&2 \end{array}\displaystyle \right ),\qquad D=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 2 &0.5&0.5&0.5\\ 1&1&1&1\\ 0.5&0&2&0.5\\ 0&1&1&2 \end{array}\displaystyle \right ). \end{aligned}$$
(i) It is easy to calculate that
\(\rho(A\circ B)=5.4983\). By inequalities (
2.1) and (
2.2), we have
$$ \rho(A\circ B)\leq\max_{1\leq i\leq4 }\bigl\{ a_{ii}b_{ii}+ \bigl(\rho(A)-a_{ii}\bigr) \bigl(\rho(B)-b_{ii}\bigr)\bigr\} =16.3949, $$
and
$$\rho(A\circ B)\leq11.6478. $$
By inequality (
2.3), we have
$$\rho(A\circ B)\leq\rho(AB)=19.05. $$
(ii) From calculation, we get
\(\rho(A\circ B\circ C)=12.0014\). By inequalities (
2.1) and (
2.2), we have
$$ \rho(A\circ B\circ C)\leq\max_{1\leq i\leq4 }\bigl\{ a_{ii}b_{ii}c_{ii}+ \bigl(\rho(A)-a_{ii}\bigr) \bigl(\rho(B)-b_{ii}\bigr) \bigl( \rho(C)-c_{ii}\bigr)\bigr\} =20.8846, $$
and
$$\rho(A\circ B\circ C) \leq17.8268. $$
By inequality (
2.3), we have
$$\rho(A\circ B\circ C)\leq\rho(ABC)=88.5. $$
(iii) Let
\(A\circ B\circ C \circ D=G=(g_{ij})\). Then
$$ G=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 16 &0&0&1\\ 0&0.2&0.5&0.5\\ 0&0&24&0.075\\ 0&0.5&0&16 \end{array}\displaystyle \right ),\qquad ABCD=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 117.25 &78.75&155.75&126 \\ 34.3375 & 23.0625 & 45.6125 & 36.9\\ 75.375 & 50.625& 100.125 & 81\\ 92.125 & 61.875& 122.375 & 99 \end{array}\displaystyle \right ). $$
It is easy to calculate that
\(\rho(G)=24.0001\). By inequalities (
2.1) and (
2.2), we have
$$ \rho(G)\leq\max_{1\leq i\leq4} \bigl\{ a_{ii}b_{ii}c_{ii}d_{ii}+ \bigl(\rho(A)-a_{ii}\bigr) \bigl(\rho(B)-b_{ii}\bigr) \bigl( \rho (C)-c_{ii}\bigr) \bigl(\rho(D)-d_{ii}\bigr)\bigr\} =36.6608 $$
and
$$\begin{aligned} \rho(G) \leq&\max_{i\neq j} \frac{1}{2}\bigl\{ g_{ii}+g_{jj}+\bigl[(g_{ii}-g_{jj})^{2}+4 \bigl(\rho(A)-a_{ii}\bigr) \bigl(\rho (B)-b_{ii}\bigr) \bigl( \rho(C)-c_{ii}\bigr) \\ &{}\times\bigl(\rho(D)-d_{ii}\bigr) \bigl(\rho(A)-a_{jj} \bigr) \bigl(\rho(B)-b_{jj}\bigr) \bigl(\rho(C)-c_{jj}\bigr) \bigl(\rho (D)-d_{jj}\bigr)\bigr]^{\frac{1}{2}}\bigr\} \\ =&32.4451. \end{aligned}$$
By inequality (
2.3), we have
\(\rho(G)\leq\rho(ABCD)=339.44\).
Next, we will give some other inequalities for
\(\rho(A_{1}\circ A_{2}\circ \cdots\circ A_{k})\). For
\(A_{1}\geq0\), write
\(L_{1}=A_{1}-\operatorname{diag}(a_{11}, \ldots,a_{nn})\). We denote
\(J_{A_{1}}=D_{1}^{-1}L_{1} \) with
\(D_{1}=\operatorname{diag}(d_{ii})\), where
$$d_{ii}= \textstyle\begin{cases} a_{ii},& \mbox{if } a_{ii}\neq0, \\ 1,& \mbox{if } a_{ii}= 0. \end{cases} $$
Then
\(J_{A_{1}}\) is nonnegative.
For
\(A_{2}\geq0\), let
\(D_{2}=\operatorname{diag}(s_{ii}), \ldots\) , for
\(A_{k} \geq0\), let
\(D_{k}=\operatorname{diag}(t_{ii})\) with
$$\begin{aligned}& s_{ii}= \textstyle\begin{cases} b_{ii},& \mbox{if } b_{ii}\neq0, \\ 1,& \mbox{if } b_{ii}= 0, \end{cases}\displaystyle \\& \ldots, \\& t_{ii}= \textstyle\begin{cases} k_{ii},& \mbox{if } k_{ii}\neq0, \\ 1,& \mbox{if } k_{ii}= 0, \end{cases}\displaystyle \end{aligned}$$
respectively. Then the nonnegative matrix
\(J_{A_{2}}, \ldots, J_{A_{k}}\) can be similarly defined.
Setting
\(k=2\) in Theorem
2.3, we have the following corollary.
Setting
\(k=2\) in Theorem
2.4, we have the following corollary.
We next give a comparison between the upper bound in (
2.4) and the upper bound in (
2.5). Without loss of generality, for
\(i\neq j\), assume that
$$\begin{aligned}& a_{ii}b_{ii}\cdots k_{ii}+d_{ii}s_{ii} \cdots t_{ii} \rho (J_{A_{1}})\rho(J_{A_{2}})\cdots \rho(J_{A_{k}}) \\& \quad \geq a_{jj}b_{jj} \cdots k_{jj}+d_{jj}s_{jj} \cdots t_{jj} \rho (J_{A_{1}})\rho(J_{A_{2}})\cdots \rho(J_{A_{k}}). \end{aligned}$$
Let
\(\gamma=a_{ii}b_{ii}\cdots k_{ii}+a_{jj}b_{jj}\cdots k_{jj}\). From (
2.5), we have
$$\begin{aligned}& a_{ii}b_{ii}\cdots k_{ii}+a_{jj}b_{jj} \cdots k_{jj}+ \bigl[(a_{ii}b_{ii}\cdots k_{ii}-a_{jj}b_{jj}\cdots k_{jj})^{2} \\& \qquad {}+4(d_{ii}s_{ii}\cdots t_{ii}) (d_{jj}s_{jj}\cdots t_{jj}) \bigl(\rho ^{2}(J_{A_{1}})\rho^{2}(J_{A_{2}})\cdots \rho^{2}(J_{A_{k}})\bigr) \bigr]^{\frac {1}{2}} \\& \quad \leq\gamma+ \bigl\{ (a_{ii}b_{ii}\cdots k_{ii}-a_{jj}b_{jj}\cdots k_{jj})^{2}+ 4d_{ii}s_{ii}\cdots t_{ii}\rho(J_{A_{1}}) \rho(J_{A_{2}})\cdots \rho(J_{A_{k}}) \\& \qquad {}\times\bigl[d_{ii}s_{ii}\cdots t_{ii} \rho(J_{A_{1}})\rho (J_{A_{2}})\cdots\rho(J_{A_{k}})+a_{ii}b_{ii} \cdots k_{ii}-a_{jj}b_{jj}\cdots k_{jj} \bigr]\bigr\} ^{\frac{1}{2}} \\& \quad =\gamma+\bigl[ \bigl(a_{ii}b_{ii}\cdots k_{ii}-a_{jj}b_{jj}\cdots k_{jj} +2d_{ii}s_{ii}\cdots t_{ii}\rho(J_{A_{1}}) \rho(J_{A_{2}})\cdots\rho (J_{A_{k}})\bigr)^{2} \bigr]^{\frac{1}{2}} \\& \quad =2a_{ii}b_{ii}\cdots k_{ii}+2d_{ii}s_{ii} \cdots t_{ii}\rho (J_{A_{1}})\rho(J_{A_{2}})\cdots \rho(J_{A_{k}}). \end{aligned}$$
Thus, from (
2.5) and the above inequality, we have
$$\begin{aligned}& \rho(A_{1}\circ A_{2}\circ\cdots\circ A_{k}) \\& \quad \leq\max_{ i\neq j} \frac{1}{2}\bigl\{ a_{ii}b_{ii}\cdots k_{ii}+a_{jj}b_{jj} \cdots k_{jj}+ \bigl[(a_{ii}b_{ii}\cdots k_{ii}-a_{jj}b_{jj}\cdots k_{jj})^{2} \\& \qquad {}+4(d_{ii}s_{ii}\cdots t_{ii}) (d_{jj}s_{jj}\cdots t_{jj}) \bigl( \rho^{2}(J_{A_{1}})\rho^{2}(J_{A_{2}})\cdots \rho^{2}(J_{A_{k}})\bigr)\bigr]^{\frac{1}{2}}\bigr\} \\& \quad \leq\max_{ 1 \leq i\leq n} \frac{1}{2}\bigl(2a_{ii}b_{ii} \cdots k_{ii}+2d_{ii}s_{ii}\cdots t_{ii} \rho(J_{A_{1}})\rho(J_{A_{2}})\cdots\rho (J_{A_{k}})\bigr) \\& \quad =\max_{ 1 \leq i \leq n} \bigl(a_{ii}b_{ii}\cdots k_{ii}+d_{ii}s_{ii}\cdots t_{ii} \rho(J_{A_{1}})\rho(J_{A_{2}})\cdots\rho (J_{A_{k}})\bigr). \end{aligned}$$
Hence, the upper bound (
2.5) is better than bound (
2.4). Here too, we could not give the comparison between (
2.4) and (
2.3) or (
2.5) and (
2.3). Next, we give an example which shows that the results obtained in Theorems
2.3 and
2.4 are better than inequalities (
2.3).
Let
$$\begin{aligned}& A=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 2 &0&1&1\\ 1&4&0.5&0.5\\ 1&0&3&0.5\\ 0.5&1&1&2 \end{array}\displaystyle \right ), \qquad B=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 2 &0.5&0.5&0.5\\ 1&1&1&1\\ 0.5&0&2&0.5\\ 0&1&1&2 \end{array}\displaystyle \right ), \\& C=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 4 &1&1&1\\ 2&2&1&1\\ 0&2&2&1\\ 1&1&1&1 \end{array}\displaystyle \right ),\qquad D=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 1 &0.5&2&0.5\\ 0.5&1&0.5&0\\ 0&0.5&1&0.5\\ 0&1&0.5&1 \end{array}\displaystyle \right ). \end{aligned}$$
Let
\(A\circ B\circ C \circ D=P=(p_{ij})\). Then
$$ P=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 16 &0&1&0.25\\ 1&8&0.25&0\\ 0&0&12&0.075\\ 0&1&0.5&4 \end{array}\displaystyle \right ),\qquad ABCD=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 35.5 &55.75 & 86 & 35.75\\ 57.75 & 91.875 & 139 & 57.875\\ 30.25 & 57.25 & 78 & 34.75\\ 34.875 &64.125 & 87.5 &38.875 \end{array}\displaystyle \right ). $$
It is easy to calculate that
\(\rho(P)=16.0028\). By inequalities (
2.4) and (
2.5), we have
$$ \rho(P)\leq\max_{1\leq i\leq4} \bigl\{ p_{ii}\bigl(1+ \rho(J_{A})\rho(J_{B})\rho(J_{C}) \rho(J_{D})\bigr)\bigr\} =36.2262 $$
and
$$\begin{aligned} \rho(P) \leq&\max_{i\neq j} \frac{1}{2}\bigl\{ p_{ii}+p_{jj}+\bigl[(p_{ii}-p_{jj})^{2}+4p_{ii}p_{jj} \rho ^{2}(J_{A})\rho^{2}(J_{B}) \rho^{2}(J_{C})\rho^{2}(J_{D}) \bigr]^{\frac{1}{2}}\bigr\} \\ =&29.6605. \end{aligned}$$
By inequality (
2.3) and Lemma
2.1, we have
\(\rho(P)\leq91.875\).
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