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# Monotonicity of the number of positive entries in nonnegative matrix powers

*Journal of Inequalities and Applications*
**volumeÂ 2018**, ArticleÂ number:Â 255 (2018)

## Abstract

Let *A* be a nonnegative matrix of order *n* and \(f(A)\) denote the number of positive entries in *A*. We prove that if \(f(A)\leq3\) or \(f(A)\geq n^{2}-2n+2\), then the sequence \(\{f(A^{k})\}_{k=1}^{\infty}\) is monotonic for positive integers *k*.

## 1 Introduction

A matrix is *nonnegative (positive)* if all of its entries are nonnegative (positive) real numbers. Nonnegative matrices have many attractive properties and are important in a variety of applications [1, 2]. For two nonnegative matrices *A* and *B* of the same size, the notation \(A\geq B\) or \(B\leq A\) means that \(A-B\) is nonnegative.

A *sign pattern* is a matrix whose entries are from the set \(\{ +, -, 0\}\). In a talk at the 12th ILAS conference (Regina, Canada, June 26â€“29, 2005), Professor Xingzhi Zhan posed the following problem.

### Problem

([4], p. 233)

*Characterize those sign patterns of square nonnegative matrices*
*A*
*such that the sequence*
\(\{f(A^{k})\}_{k=1}^{\infty}\)
*is nondecreasing*.

A nonnegative square matrix *A* is said to be *primitive* if there exists a positive integer *k* such that \(A^{k}\) is positive. If we denote by \(f(A)\) the number of positive entries in *A*, it seems that the sequence \(\{f(A^{k})\}_{k=1}^{\infty}\) is increasing for any primitive matrix *A*. However, Å idÃ¡k [3] observed that there is a primitive matrix *A* of order 9 satisfying \(f(A)=18>f(A^{2})=16\). This is the motivation for us to investigate the nonnegative matrix *A* such that \(\{f(A^{k})\}_{k=1}^{\infty}\) is monotonic. It is reasonable to expect that the sequence will be monotonic when \(f(A)\) is too small or too large.

Since the value of each positive entry in *A* does not affect \(f(A^{k})\) for all positive integers *k*, it suffices to consider the 0â€“1 matrix, i.e., the matrix whose entries are either 0 or 1. Denote by \(E_{ij}\) the matrix with its entry in the *i*th row and *j*th column being 1 and with all other entries being 0. For simplicity we use 0 to denote the zero matrix whose size will be clear from the context.

## 2 Main results

Let *A* be a nonnegative square matrix. We will use the fact that if \(A^{2}\geq A\ (A^{2}\leq A)\), then \(A^{k+1}\geq A^{k}\ (A^{k+1}\leq A^{k})\) for all positive integers *k* and thus \(\{f(A^{k})\}_{k=1}^{\infty}\) is increasing (decreasing).

### Theorem 1

*Let*
*A*
*be a* 0*â€“*1 *matrix of order*
*n*. *If*
\(f(A)\leq2\), *then the sequence*
\(\{f(A^{k})\}_{k=1}^{\infty}\)
*is decreasing*.

### Proof

The case \(f(A)=0\) is trivial.

If \(f(A)=1\), then \(A=E_{ij}\), \(1\leq i,j\leq n\). Thus, for \(k=2,3,\ldots\)â€‰,

which implies that \(\{f(A^{k})\}_{k=1}^{\infty}\) is decreasing. Next suppose \(f(A)=2\).

Since \(\{f(A^{k})\}_{k=1}^{\infty}\) is invariant under permutation similarity or transpose of *A*, it suffices to consider the following cases.

(1) \(A=E_{11}+E_{22}\). Then \(A^{2}=A\).

(2) \(A=E_{11}+E_{12}\). Then \(A^{2}=A\).

(3) \(A=E_{11}+E_{23}\). Then \(A^{2}=E_{11}\leq A\).

(4) \(A=E_{12}+E_{13}\). Then \(A^{2}=0\).

(5) \(A=E_{12}+E_{21}\). Then \(A^{k}=E_{11}+E_{22}\) for all even *k*, \(A^{k}=A\) for all odd *k*.

(6) \(A=E_{12}+E_{23}\). Then \(A^{2}=E_{13}\), \(A^{3}=0\).

(7) \(A=E_{12}+E_{34}\). Then \(A^{2}=0\).

It can be seen that in each case \(\{f(A^{k})\}_{k=1}^{\infty}\) is decreasing. This completes the proof.â€ƒâ–¡

### Theorem 2

*Let*
*A*
*be a* 0*â€“*1 *matrix of order*
*n*. *If*
\(f(A)=3\), *then the sequence*
\(\{f(A^{k})\}_{k=1}^{\infty}\)
*is monotonic*.

### Proof

Under permutation similarity and transpose, it suffices to consider the following cases.

(1) \(A=E_{11}+E_{22}+E_{33}\). Then \(A^{2}=A\).

(2) \(A=E_{11}+E_{22}+E_{12}\). Then \(A^{2}=A+E_{12}\geq A\).

(3) \(A=E_{11}+E_{22}+E_{13}\). Then \(A^{2}=A\).

(4) \(A=E_{11}+E_{22}+E_{34}\). Then \(A^{2}=E_{11}+E_{22}\leq A\).

(5) \(A=E_{11}+E_{12}+E_{13}\). Then \(A^{2}=A\).

(6) \(A=E_{11}+E_{12}+E_{21}\). Then \(A^{2}=A+E_{11}+E_{22}\geq A\).

(7) \(A=E_{11}+E_{12}+E_{31}\). Then \(A^{2}=A+E_{32}\geq A\).

(8) \(A=E_{11}+E_{12}+E_{23}\). Then \(A^{k}=E_{11}+E_{12}+E_{13}\) for all \(k\geq2\).

(9) \(A=E_{11}+E_{12}+E_{32}\). Then \(A^{2}=E_{11}+E_{12}\leq A\).

(10) \(A=E_{11}+E_{12}+E_{34}\). Then \(A^{2}=E_{11}+E_{12}\leq A\).

(11) \(A=E_{11}+E_{23}+E_{24}\). Then \(A^{2}=E_{11}\leq A\).

(12) \(A=E_{11}+E_{23}+E_{32}\). Then \(A^{k}=E_{11}+E_{22}+E_{33}\) for all even *k*, \(A^{k}=A\) for all odd *k*.

(13) \(A=E_{11}+E_{23}+E_{34}\). Then \(A^{2}=E_{11}+E_{24}\), \(A^{k}=E_{11}\) for all \(k\geq3\).

(14) \(A=E_{11}+E_{23}+E_{45}\). Then \(A^{2}=E_{11}\leq A\).

(15) \(A=E_{12}+E_{13}+E_{14}\). Then \(A^{2}=0\).

(16) \(A=E_{12}+E_{13}+E_{21}\). Then \(A^{k}=E_{11}+E_{22}+E_{23}\) for all even *k*, \(A^{k}=A\) for all odd *k*.

(17) \(A=E_{12}+E_{13}+E_{41}\). Then \(A^{2}=E_{42}+E_{43}\), \(A^{3}=0\).

(18) \(A=E_{12}+E_{13}+E_{23}\). Then \(A^{2}=E_{13}\leq A\).

(19) \(A=E_{12}+E_{13}+E_{24}\). Then \(A^{2}=E_{14}\), \(A^{3}=0\).

(20) \(A=E_{12}+E_{13}+E_{42}\). Then \(A^{2}=0\).

(21) \(A=E_{12}+E_{13}+E_{45}\). Then \(A^{2}=0\).

(22) \(A=E_{12}+E_{21}+E_{34}\). Then \(A^{k}=E_{11}+E_{22}\) for all even *k*, \(A^{k}=E_{12}+E_{21}\) for all odd \(k\geq3\).

(23) \(A=E_{12}+E_{23}+E_{31}\). Then

(24) \(A=E_{12}+E_{23}+E_{34}\). Then \(A^{2}=E_{13}+E_{24}\), \(A^{3}=E_{14}\), \(A^{4}=0\).

(25) \(A=E_{12}+E_{23}+E_{45}\). Then \(A^{2}=E_{13}\), \(A^{3}=0\).

(26) \(A=E_{12}+E_{34}+E_{56}\). Then \(A^{2}=0\).

Since in each case \(\{f(A^{k})\}_{k=1}^{\infty}\) is either increasing or decreasing, this completes the proof.â€ƒâ–¡

### Corollary 3

*Let*
*A*
*be a* 0*â€“*1 *matrix of order* 2. *Then the sequence*
\(\{f(A^{k})\}_{k=1}^{\infty}\)
*is monotonic*.

### Remark

When *A* is of order \(n\geq3\) with \(f(A)=4\), the following example shows that \(\{f(A^{k})\}_{k=1}^{\infty}\) may not be monotonic. Consider

Direct computation shows that

Thus \(f(A)=4< f(A^{2})=5>f(A^{3})=4\).

On the one hand, Theorems 1 and 2 show that \(\{f(A^{k})\}_{k=1}^{\infty}\) is monotonic when \(f(A)\leq3\). On the other hand, \(\{f(A^{k})\}_{k=1}^{\infty}\) is expected to be also monotonic when \(f(A)\) is large enough. Next we discuss the number of positive entries that *A* has to guarantee the sequence increasing.

The *permanent* of a matrix \(A=(a_{ij})_{n\times n}\) is defined as

where \(S_{n}\) is the set of permutations of the integers \(1,2,\ldots,n\). First we have the following important fact.

### Lemma 4

*Let*
*A*
*be a* 0*â€“*1 *matrix of order*
*n*. *If*
\(\operatorname {per}A>0\), *then the sequence*
\(\{f(A^{k})\}_{k=1}^{\infty}\)
*is increasing*.

### Proof

Since *A* is a 0â€“1 matrix with \(\operatorname {per}A>0\), there exists a permutation matrix *P* such that \(A\geq P\). Now let \(A=P+B\), where *B* is also a 0â€“1 matrix. Then \(A^{k+1}=A\cdot A^{k}=(P+B)A^{k}=P\cdot A^{k}+B\cdot A^{k}\geq P\cdot A^{k}\) for all positive integers *k*. Thus \(f(A^{k+1})\geq f(P\cdot A^{k})=f(A^{k})\), which implies that \(\{ f(A^{k})\}_{k=1}^{\infty}\) is increasing.â€ƒâ–¡

### Theorem 5

*Let*
*A*
*be a* 0*â€“*1 *matrix of order*
*n*. *If*
\(f(A)\geq n^{2}-2n+2\), *then the sequence*
\(\{f(A^{k})\}_{k=1}^{\infty}\)
*is increasing*.

### Proof

First if \(\operatorname {per}A>0\), by Lemma 4, \(\{f(A^{k})\} _{k=1}^{\infty}\) is increasing.

Next suppose \(\operatorname {per}A=0\). Then by the Frobeniusâ€“KÃ¶nig theorem [4, p. 46], *A* has an \(r\times s\) zero submatrix with \(r+s=n+1\). Since \(f(A)\geq n^{2}-2n+2\), *A* has at most \(2n-2\) zero entries. Thus \(rs\leq2n-2\). It can be seen that *r* and *s* must be one of the following solutions.

(1) \(r=1\), \(s=n\);

(2) \(r=n\), \(s=1\);

(3) \(r=2\), \(s=n-1\);

(4) \(r=n-1\), \(s=2\).

If \(r=1\), \(s=n\) or \(r=n\), \(s=1\), i.e., *A* has a zero row or a zero column, then *A* is permutation similar to a matrix of the form

or its transpose, where *B* is of order \(n-1\) and *C* is a column vector. Since *A* has at most \(2n-2\) zero entries, *B* has at most \(n-2\) zero entries. Then there exists a permutation matrix *Q* of order \(n-1\) such that \(B\geq Q\). Note that

Thus

for all positive integers *k*, which implies that \(\{f(A^{k})\} _{k=1}^{\infty}\) is increasing.

If \(r=2\), \(s=n-1\) or \(r=n-1\), \(s=2\), then *A* is permutation similar to one of the matrices \(A_{1}, A_{2}, A_{1}^{T}, A_{2}^{T}\), where

Direct computation shows that \(A_{1}^{2}\geq A_{1}, A_{2}^{2}\geq A_{2}\). Thus \(\{ f(A^{k})\}_{k=1}^{\infty}\) is increasing. This completes the proof.â€ƒâ–¡

### Remark

When \(f(A)=n^{2}-2n+1\), the following example shows that \(\{f(A^{k})\}_{k=1}^{\infty}\) may not be increasing. Consider

Direct computation shows that \(f(A)=n^{2}-2n+1>f(A^{2})=n^{2}-2n\).

## 3 Conclusion

This paper considers the number of positive entries \(f(A)\) in a nonnegative matrix *A* and deals with the question of whether the sequence \(\{f(A^{k})\} _{k=1}^{\infty}\) is monotonic. We prove that if \(f(A)\leq3\) or \(f(A)\geq n^{2}-2n+2\), then the sequence must be monotonic. Some examples show that if \(4\leq f(A)\leq n^{2}-2n+1\) when \(n\geq3\), then the sequence may not be monotonic.

## References

Bapat, R.B., Raghavan, T.E.S.: Nonnegative Matrices and Applications. Cambridge University Press, Cambridge (1997)

Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia (1994)

Å idÃ¡k, Z.: On the number of positive elements in powers of a non-negative matrix. ÄŒas. PÄ›st. Mat.

**89**, 28â€“30 (1964)Zhan, X.: Matrix Theory. Grad. Stud. Math., vol.Â 147. Amer. Math. Soc., Providence (2013)

### Acknowledgements

The author would like to express her sincere thanks to referees and the editor for their enthusiastic guidance and help.

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## Funding

This research was supported by the National Natural Science Foundation of China (Grant No. 71503166).

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Xie, Q. Monotonicity of the number of positive entries in nonnegative matrix powers.
*J Inequal Appl* **2018**, 255 (2018). https://doi.org/10.1186/s13660-018-1833-5

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DOI: https://doi.org/10.1186/s13660-018-1833-5