A note on the rank inequality for diagonally magic matrices

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R E S E A R C H

Open Access

A note on the rank inequality for

diagonally magic matrices

Duanmei Zhou

1*

_{, Qingyou Cai}

1

_{and Xiaoyan Chen}

2

*_{Correspondence:}

[email protected] 1_{College of Mathematics and}

Computer Science, Gannan Normal University, Ganzhou, 341000, People’s Republic of China Full list of author information is available at the end of the article

Abstract

We prove that a positive matrix with all permutation products equal is diagonally equivalent toJ, the all-1s matrix. Then we give a simple proof of the rank inequality for diagonally magic matrices (J. Inequal. Appl. 2015:318, 2015).

MSC: 15A03; 15A06

Keywords: rank; diagonally magic matrices; positive matrix

1 Introduction

We denote byCn×nandRn×nthe sets ofn×ncomplex matrices andn×nreal matrices, respectively. For a positive integern, letSnbe the set of alln! permutations of{, , . . . ,n}. IfA= (ai,j)∈Cn×n andσ ∈Sn, then the sequence a,σ(),a,σ(), . . . ,an,σ(n) is called the transversalofA[]. LetA∈Cn×n_{, }_≤_i

≤i≤ · · · ≤ik≤n, and ≤j≤j≤ · · · ≤js≤n. We denote by A[i,i, . . . ,ik|j,j, . . . ,js] the k×s submatrix of A that lies in the rows i,i, . . . ,ikand columnsj,j, . . . ,js. Denote byA(i,i, . . . ,ik|j,j, . . . ,js) the (n–k)×(n–s) submatrix ofAobtained by deleting the rowsi,i, . . . ,ikand columnsj,j, . . . ,js. A matrix A= (ai,j)∈Cn×nis calleddiagonally magicif

n

i= ai,σ(i)=

n

i= ai,π(i)

for allσ,π∈Sn.

Obviously, the zero matrix n×nandJ= []n×n, the matrix of all ones, are diagonally magic matrices. In [], we prove that

Bn=

⎛ ⎜ ⎜ ⎜ ⎜ ⎝

  · · · n

n+  n+  · · · n ..

. ... . .. ...

(n– )n+  (n– )n+  · · · n

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

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and the Henkel matrix

Cn=

⎛ ⎜ ⎜ ⎜ ⎜ ⎝

  · · · n   · · · n+ 

..

. ... . .. ... n n+  · · · n– 

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

are diagonally magic matrices. So, there are a lot of diagonally magic matrices. The non-negative matricesBnandCnhave been a hot research area [, ].

2 Main result

The rank inequality for diagonally magic matrices can be stated as follows.

Theorem . ([], Theorem .) Let A∈ Cn×n _{be a diagonally magic matrix}_. _Then

rank(A)≤.

There are diagonally magic matrices of ranks , , . Indeed, rank(n×n) = ,

rank([]n×n) = , andrank(Bn) =rank(Cn) = .

The purpose of this note is to give a simple proof of Theorem .. Our proof depends only on the following fact.

Theorem . Let C= (ci,j)∈Rn×nbe a positive matrix with n

i= ci,γ(i)=

n

j= cj,τ(j)

for all γ,τ ∈Sn.Then there exist positive diagonal matrices X=diag(x,x, . . . ,xn)and Y=diag(y,y, . . . ,yn)such that

C=XJY.

Proof Let Bbe a k×k submatrix of C. Then there are row and column indices α= (i,i, . . . ,ik) andβ= (j,j, . . . ,jk) such thatB=C[α|β]. Note that the union of a transver-sal ofBand a transversal of C(α|β) is a transversal ofC. Choose an arbitrary but ﬁxed transversalTof the square matrixC(α|β). For anyσ,π∈Sk,ci,jσ(), . . . ,cik,jσ(k)and the

en-tries inTconstitute a transversal ofC, whereasci,jπ(), . . . ,cik,jπ(k)and the entries inTalso

constitute a transversal ofC. Letbbe the product of the entries inT. Obviously,b> . Since

n

i= ci,γ(i)=

n

j= cj,τ(j)

for allγ,τ ∈Sn, we have

b k

t=

cit,jσ(t)=b

k

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which yields

k

t=

cit,jσ(t)=

k

t= cit,jπ(t).

Particularly, this shows that any × submatrix

B=

ci,j ci,j

ci,j ci,j

ofCsatisﬁes

ci,jci,j=ci,jci,j. ()

For anyx> , let

yj= c,j

x

()

forj= , , . . . ,nand

xi= ci, c,

x ()

fori= , , . . . ,n. According to (), (), and (), we have

ci,j= ci,c,j

c, =xiyj

for alli,j= , , . . . ,n. LetX=diag(x,x, . . . ,xn) andY=diag(y,y, . . . ,yn). Obviously,X andYare positive diagonal matrices, and we have

C=XJY.

This completes the proof.

We are now ready to present our proof of Theorem ..

Proof of Theorem. First, letAbe real. LetAbe a diagonally magic matrix. Then the ele-mentwise exponentialC=exp(A) := (ci,j)∈Rn×nis a positive matrix with all permutation products equal. Hence, by Theorem . it is diagonally equivalent toJ, the all-s matrix, that is,

ci,j=xiyj, i,j= , , . . . ,n,

for suitable positive vectors x = (x,x, . . . ,xn)T and y= (y,y, . . . ,yn)T. Hence, if q= (log(x),log(x), . . . ,log(xn))Tandr= (log(y),log(y), . . . ,log(yn))T, then

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Hence,

A=en·qT+r·eTn,

whereen= (, , . . . , 

n

)T_{. Thus,}_A_{is the sum of two matrices of rank  and, hence, at most}

of rank .

Now letAbe complex, so that

A=B+iC (B,Creal).

SinceAis a diagonally magic matrix, so areBandC. Hence, bothBandCare of the form

B=en·qT +r·eTn withq,rreal,

C=en·qT +r·eTn withq,rreal,

and hence

A=en·(q+iq)T+ (r+ir)·eTn. ()

The matrixAhas rank at most . This completes the proof.

According to (), we obtain that a diagonally magic matrixAcan be presented in the form

A=en·xT+y·eTn =

⎛ ⎜ ⎜ ⎜ ⎜ ⎝

x+y x+y · · · x+yn x+y x+y · · · x+yn

..

. ... . .. ... xn+y xn+y · · · xn+yn

⎞ ⎟ ⎟ ⎟ ⎟

⎠. ()

IfA= (ai,j)∈Cn×nis a diagonally magic matrix, for anyx∈C, let yj=a,j–x

forj= , , . . . ,nand

xi=ai,–a,+x

fori= , , . . . ,n. By () we have

ai,j=xi+yj

for alli,j= , , . . . ,n. For example,

Bn=

⎛ ⎜ ⎜ ⎜ ⎜ ⎝

  · · · n

n+  n+  · · · n ..

. ... . .. ...

(n– )n+  (n– )n+  · · · n

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= ⎛ ⎜ ⎜ ⎜ ⎜ ⎝

 +   +  · · ·  +n n+  n+  · · · n+n

..

. ... . .. ...

(n– )n+  (n– )n+  · · · (n– )n+n

⎞ ⎟ ⎟ ⎟ ⎟ ⎠ and

Cn=

⎛ ⎜ ⎜ ⎜ ⎜ ⎝

  · · · n   · · · n+ 

..

. ... . .. ... n n+  · · · n– 

⎞ ⎟ ⎟ ⎟ ⎟ ⎠= ⎛ ⎜ ⎜ ⎜ ⎜ ⎝

 +   +  · · ·  + (n– )  +   +  · · ·  + (n– )

..

. ... . .. ... n+  n+  · · · n+ (n– )

⎞ ⎟ ⎟ ⎟ ⎟ ⎠.

By () we can get the characteristic polynomial, the eigenvalues, and the eigenvectors ofA. In fact, the characteristic polynomial ofAis

pA(λ) =λn–

λ–λ

_n

i= (xi+yi)

+ n i= xi n j= yj–n

n

i= (xiyi)

. ()

From () we can see thatthe algebraic multiplicity of the eigenvalue of the diagonally magic matrixAisat least n– .

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the ﬁnal manuscript.

Author details

1_{College of Mathematics and Computer Science, Gannan Normal University, Ganzhou, 341000, People’s Republic of}

China.2_{Library, Gannan Normal University, Ganzhou, 341000, People’s Republic of China.}

Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 11501126, No. 11471122), the Youth Natural Science Foundation of Jiangxi Province (No. 20151BAB211011), the Science Foundation of Jiangxi Provincial Department of Education (No. GJJ150979), the Supporting the Development for Local Colleges and Universities Foundation of China - Applied Mathematics Innovative team building, and the Education Department of Jiangxi Province (No. 15YB106).

Received: 18 December 2015 Accepted: 5 February 2016 References

1. Zhou, D, Chen, G, Cai, Q, Chen, X: The rank inequality for diagonally magic matrices. J. Inequal. Appl.2015, 318 (2015) 2. Zhan, X: Matrix Theory. Am. Math. Soc., Providence (2013)

3. Ding, J, Rhee, N: When a matrix and its inverse are nonnegative. Mo. J. Math. Sci.26, 98-103 (2014)