Volume 2010, Article ID 254819,5pages doi:10.1155/2010/254819
Research Article
On Linear Maps Preserving g-Majorization
from
F
n
to
F
m
Ali Armandnejad and Hossein Heydari
Department of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 7713936417, Rafsanjan, Iran
Correspondence should be addressed to Ali Armandnejad,[email protected]
Received 5 October 2009; Revised 7 January 2010; Accepted 16 February 2010
Academic Editor: Jong Kim
Copyrightq2010 A. Armandnejad and H. Heydari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
LetFnandFmbe the usual spaces ofn-dimensional column andm-dimensional row vectors on
F, respectively, whereFis the field of real or complex numbers. In this paper, the relations gs-majorization, lgw-gs-majorization, and rgw-majorization are considered onFn andFm. Then linear
mapsT :Fn → Fmpreserving lgw-majorization or gs-majorization and linear mapsS:Fn → Fm,
preserving rgw-majorization are characterized.
1. Introduction
Majorization is a topic of much interest in various areas of mathematics and statistics. Ifxand
yaren-vectors of real numbers such thatxDyfor some doubly stochastic matrixD, then we say thatxisvectormajorized byy; see1. Marshall and Olkin’s text2is the standard general reference for majorization. Some kinds of majorization such as multivariate or matrix majorization were motivated by the concepts of vector majorization and were introduced in
3. LetV andWbe two vector spaces over a fieldF, and let∼be a relation on bothVandW. We say that a linear mapT :V → W, preserves the relation∼if
Tx∼Ty wheneverx∼y. 1.1
The problem of describing these preserving linear maps is one of the most studied linear preserver problems. A lot of effort has been done in 4–9and 10–12to characterize the structure of majorization preserving linear maps on certain spaces of matrices. A complex
Definition 1.1. Letxandybe two vectors inFn. It is said that
1xis gs-majorized byyif there exists ann×ng-doubly stochastic matrixDsuch that
xDy, and denoted byy gs x;
2xis lgw-majorized byyif there exists ann×ng-row stochastic matrixRsuch that
xRy, and denoted byylgwx;
3xtis rgw-majorized byytif there exists ann×ng-row stochastic matrixRsuch that
xtytR, and denoted byyt
rgw xthereztis the transpose ofz.
Linear maps fromRntoRmthat preserve left matrix majorization or weak majorization were already characterized in 10, 11. In this paper we characterize all linear maps preservingrgwfromFntoFmand all linear maps preservinglgworgsfromFntoFm.
Throughout this paper, the standard bases ofFn and Fm are denoted by {e1, . . . , en}
and{1, . . . , m}, respectively. The notation trxis used for the sum of the components of a vectorx∈Fnorx∈Fn. The vector space of alln×mcomplex matrices is denoted byMn,m.
The notationsx1/x2/· · ·/xn andy1 | y2 | · · · | ymare used for the n×m matrix with
rowsx1, x2, . . . , xn ∈ Fm and columnsy1, y2, . . . , ym ∈ Fn. The sets of g-row and g-column stochasticm×nmatrices are denoted byGRm,nandGCm,n, respectively. The set of g-doubly
stochasticn×nmatrices is denoted byGDn. The symbolJnis used for then×nmatrix with all
entries equal to one. The notationTis used for the matrix representation of the linear map
T :V → Wwith respect to the standard bases ofV andWwhereV, W∈ {Fn,Fm,Fn,Fm}.
2. Main Results
In this section we state some preliminary lemmas to describe the linear maps preservingrgw fromFntoFmand the linear maps preservinglgworgsfromFntoFm.
Lemma 2.1. LetT :Fn → Fmbe a linear map. ThenT preserves the subspace{x∈Fn : trx 0}
if and only ifT∈GRm,n.
Proof. LetB bij : T. Assume thatBe λefor someλ ∈ F. Ifx ∈ Fn and trx 0,
then 0 xe xλe xBe xBe trxB trTx, so T preserves the subspace
{x∈Fn : trx 0}. Conversely, assume thatT preserves the subspace{x∈Fn : trx 0}.
Then trT1−i tr1−iB 0 for every i1 ≤ i ≤ n. ThereforeBe λe where
λnk1b1knk1bikfor everyi1≤i≤n.
The following lemma gives an equivalent condition forrgwonFm.
Lemma 2.2see4, Lemma 2.2. Letx, y∈Fnand letx /0. Thenxrgwyif and only iftrx
try.
The following theorem characterizes all linear maps which preservergw fromFn to
Fm. It is clear that everyT :F1 → Fmpreservesrgw, so assume thatn≥2.
Theorem 2.3. A nonzero linear mapT :Fn → Fmpreservesrgw if and only ifT∈GRm,nand
Proof. PutB : T. LetBe λefor someλ∈F. If{x ∈Fn :xB 0} {x∈Fn : trx 0}
it is clear thatT preservesrgw. If{x ∈ Fn : xB 0} {0},xrgw yandx /0 thenTx /0 and byLemma 2.2, trx try. So trx−y 0 and hence trTx−y 0 byLemma 2.1. ThereforeTx rgw TybyLemma 2.2and soTpreservesrgw. Now, we prove the necessity of the conditions. LetT :Fn → Fmbe a linear preserver ofrgw. If trx 0, thenxrgw 0 by Lemma 2.2. SoTx rgw T00 and hence trTx 0 byLemma 2.2. ThereforeTpreserves the subspace{x∈Fn: trx 0}and soB∈GRm,nbyLemma 2.1. If{x∈Fn:xB0}/{0}, then
there exists a nonzero vectora∈Fnsuch thatTaaB0. If tra δ /0 thenargw δjfor
everyj 1 ≤j ≤n, byLemma 2.2. ThenTa0 rgw δTj for everyj 1 ≤j ≤nand hence
T 0 which is a contradiction. Therefore tra 0 and henceargw 1−jfor everyj 1≤
j≤n, byLemma 2.2. ThenTa0 rgw T1−jand soT1 Tjfor everyj 1≤j ≤n. Put
b:T11B. ThusB b/· · ·/band hence{x∈Fn:xB0}{x∈Fn: trx 0}.
We use the following lemmas to find the structure of linear preservers of lgw-majorization.
Remark 2.4see7, Lemma 2.2. Ifx /∈Span{e}, thenxlgwy, for ally∈Fn.
Lemma 2.5. Let T : Fn → Fm be a linear map. If x /∈Span{e} implies Tx /∈Span{e}, then T preserveslgw.
Proof. Letx, y ∈Fnandxlgw y. Ifx∈Span{e}theny xand it is clear thatTx lgwTy. If x /∈Span{e} so Tx /∈Span{e} by the hypothesis and hence Txlgw Ty, by Remark 2.4. ThereforeT preserveslgw.
Lemma 2.6. LetT :Fn → Fmbe a nonzero singular linear map. ThenT preserveslgwif and only ifKerT Span{e}ande /∈ImT.
Proof. LetT be a linear preserver oflgw. Ifx ∈ KerTandx /∈Span{e}, thenTx 0 and
xlgwy, for ally ∈ Fn byRemark 2.4. SoTy 0, for all y ∈ Fn, which is a contradiction. Therefore KerT ⊂ Span{e} and since KerT/{0}, KerT Span{e}. Ife ∈ ImT, then there existsx∈Fnsuch thatTx eandx /∈Span{e}. Thereforexlgwy, for ally∈Fn, and henceTyefor ally∈Fn, which is a contradiction. Soe /∈ImT. The converse follows from Lemma 2.5.
Proposition 2.7. LetT :Fn → Fmbe a nonzero linear preserver oflgw. Thenn≤m.
Proof. If T is injective, then n ≤ m. If T is not injective, we obtain KerT Span{e} by Lemma 2.6ande /∈ImT. Thereforen≤m, by the rank and nullity theorem.
Theorem 2.8. LetT :Fn → Fmbe a nonzero linear map andA: T. ThenT preserveslgwif and only if one of the following holds:
i{x:Ax∈Span{e}}{0},
ii A∈Span{GRn,m}and{x:Ax∈Span{e}}Span{e}.
Proof. If i orii holds, it is easy to show thatT preserves lgw by Lemmas 2.5and 2.6. Conversely, assume thatT preserveslgw. Ifidoes not hold, we show thatiiholds. Since
i does not hold, there exists a nonzero vectorb ∈ Fn such thatTb Ab μefor some
and henceT 0, which is a contradiction. Thenb λefor some nonzeroλ ∈F, and hence
Ae μ/λe. Therefore,A∈Span{GRn,m}and{x:Ax∈Span{e}}Span{e}.
The following examples show thatProposition 2.7does not hold forgsorrgw.
Example 2.9. For any positive integern, the linear mapT : Fn → Fdefined byTx trx, preservesgs.
Example 2.10. The linear mapT :F3 → F2defined byTxxB, whereB 1 1 1
0 0 0 t
, preserves rgw-majorization.
We use the following statements to find the structure of linear preservers of gs-majorization.
Lemma 2.11see6, Proposition 2.1. Let x and y be two distinct vectors inFn. Thenygs xif and only ify /∈Span{e}andtrx try.
Lemma 2.12. If a linear mapT :Fn → Fmpreservesgs, thenT∈Span{GCm,n}.
Proof. LetA: T. For everyi, j 1≤ i /j ≤n, it is clear thatei−ejgs 0 byLemma 2.11. ThenAei−ejgs 0 and hence there existsD∈GDmsuch thatDAei−ej 0. SoJmAei−ej JmDAei−Aej 0 and thereforeA∈Span{GCm,n}.
Theorem 2.13. LetT : Fn → Fm be a linear map. ThenT preservesgsif and only if one of the following holds:
ithere exists somea∈Fmsuch thatTxtrxa, for allx∈Fn,
iiλT∈GRm,n∩Span{GCm,n}for some0/λ∈FandKerT⊂Span{e},
iii T∈Span{GCm,n}ande∈ImT.
Proof. LetA: T. Assume thatT preservesgs. SoA∈Span{GCm,n}byLemma 2.12. Now,
we consider two cases.
Case 1. Suppose there exists b ∈ Fn\Span{e}such thatTb Ab λefor someλ ∈ F. If trb 0, then 0trbe Jmb JmAb JmAb JmTb Jmλe. Soλ 0 and hence
Ab0. For everyi, j 1≤i /j≤n,bgsei−ejbyLemma 2.11. Then 0Abgs Aei−ej and henceAei Aej, for alli, j 1 ≤ i, j ≤ n. ThenA a| · · · |a, for somea∈ Fmand
henceTx trxafor allx ∈ Fn. If trb δ /0, consider the basis{δe1, . . . , δen}forFn.
For everyi1 ≤ i ≤ n,bgsδei, byLemma 2.11. ConsequentlyTei λ/δe for every
i1≤i≤nand henceTx trxafor allx∈Fn, wherea λ/δe. Therefore,iholds in this case.
Case 2. Assume thatx∈Span{e}impliesTx∈Span{e}. Sincee1 gs ei, we haveTe1gs Tei for every i1 ≤ i ≤ n. Thus it follows that trAi trTei trTe1 trA1for every
i1 ≤ i ≤ n, whereAi is theith column ofAand henceA ∈ Span{GCm,n}. Ife ∈ ImA,
then there exists 0/λ ∈ Fsuch thatAλe eand henceλA ∈ GRm,n∩Span{GCm,n}. By
Conversly, if i or iii holds it is easy to show that T preserves gs-majorization. Suppose thatiiholds. Then there existsz∈Span{e}such thatTze. Assume thatxgs y. IfTx∈Span{e}thenTxgsTybyLemma 2.11. IfTx∈Span{e}, then there existsμ∈Fsuch thatTxμeand henceTx−μz 0. Therefore,x−μz∈Span{e}, and hencex∈Span{e}. Thenxyand henceTpreserves gs-majorization.
Corollary 2.14. IfT :Fn → FmpreservesgsandrankT>1thenn≤m.
Proof. IfT is injective it is clear thatn ≤ m. Assume that T is not injective, so there exists a nonzero vectorb ∈ Fn such thatTb 0. Ifb /∈Span{e}, then by Case1 in the proof of Theorem 2.13,Txtrxafor somea∈Fm. Therefore, rankT≤1, which is a contradiction. Sob ∈Span{e}and hence KerT Span{e}. It is clear thate /∈ImT, from which and the rank and nullity theorem, we obtainn≤m, completing the proof.
Acknowledgment
The authors would like to sincerely thank the referees for their constructive comments and suggestions which made some of the proofs simpler and clearer. This work has been supported by Vali-e-Asr university of Rafsanjan, grant no. 2740.
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