- Research Article
- Open Access
On Linear Maps Preserving g-Majorization from to
© A. Armandnejad and H. Heydari. 2010
- Received: 5 October 2009
- Accepted: 16 February 2010
- Published: 1 March 2010
Let and be the usual spaces of n-dimensional column and m-dimensional row vectors on , respectively, where is the field of real or complex numbers. In this paper, the relations gs-majorization, lgw-majorization, and rgw-majorization are considered on and . Then linear maps preserving lgw-majorization or gs-majorization and linear maps , preserving rgw-majorization are characterized.
- Real Number
- Vector Space
- General Reference
- Generalize Majorization
- Complex Matrix
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 [4–9] and [10–12] to characterize the structure of majorization preserving linear maps on certain spaces of matrices. A complex matrix is said to be g-row (or g-column) stochastic, if (or ), where (or ). A complex matrix is said to be g-doubly stochastic if it is both g-row and g-column stochastic. The notaions of generalized majorization (g-majorization) were motivated by the matrix majorization and were introduced in [4–6] as follows.
Let and be two vectors in . It is said that
(1) is gs-majorized by if there exists an g-doubly stochastic matrix such that , and denoted by ;
(2) is lgw-majorized by if there exists an g-row stochastic matrix such that , and denoted by ;
(3) is rgw-majorized by if there exists an g-row stochastic matrix such that , and denoted by (here is the transpose of ).
Linear maps from to that preserve left matrix majorization or weak majorization were already characterized in [10, 11]. In this paper we characterize all linear maps preserving from to and all linear maps preserving or from to .
Throughout this paper, the standard bases of and are denoted by and , respectively. The notation is used for the sum of the components of a vector or . The vector space of all complex matrices is denoted by . The notations and are used for the matrix with rows and columns . The sets of g-row and g-column stochastic matrices are denoted by and , respectively. The set of g-doubly stochastic matrices is denoted by . The symbol is used for the matrix with all entries equal to one. The notation is used for the matrix representation of the linear map with respect to the standard bases of and where .
In this section we state some preliminary lemmas to describe the linear maps preserving from to and the linear maps preserving or from to .
Let be a linear map. Then preserves the subspace if and only if .
Let . Assume that for some . If and , then , so preserves the subspace . Conversely, assume that preserves the subspace . Then for every . Therefore where for every .
The following lemma gives an equivalent condition for on .
Lemma 2.2 (see [4, ]).
Let and let . Then if and only if .
The following theorem characterizes all linear maps which preserve from to . It is clear that every preserves , so assume that .
A nonzero linear map preserves if and only if and .
Put . Let for some . If it is clear that preserves . If , and then and by Lemma 2.2, . So and hence by Lemma 2.1. Therefore by Lemma 2.2 and so preserves . Now, we prove the necessity of the conditions. Let be a linear preserver of . If , then by Lemma 2.2. So and hence by Lemma 2.2. Therefore preserves the subspace and so by Lemma 2.1. If , then there exists a nonzero vector such that . If then for every , by Lemma 2.2. Then for every and hence which is a contradiction. Therefore and hence for every , by Lemma 2.2. Then and so for every . Put . Thus and hence .
We use the following lemmas to find the structure of linear preservers of lgw-majorization.
Remark 2.4 (see [7, ]).
If , then , for all .
Let be a linear map. If implies , then T preserves .
Let and . If then and it is clear that . If so by the hypothesis and hence , by Remark 2.4. Therefore preserves .
Let be a nonzero singular linear map. Then preserves if and only if and .
Let be a linear preserver of . If and , then and , for all by Remark 2.4. So , for all , which is a contradiction. Therefore and since , . If , then there exists such that and . Therefore , for all , and hence for all , which is a contradiction. So . The converse follows from Lemma 2.5.
Let be a nonzero linear preserver of . Then .
If is injective, then . If is not injective, we obtain by Lemma 2.6 and . Therefore , by the rank and nullity theorem.
Let be a nonzero linear map and . Then preserves if and only if one of the following holds:
(ii) and .
If (i) or (ii) holds, it is easy to show that preserves by Lemmas 2.5 and 2.6. Conversely, assume that preserves . If (i) does not hold, we show that (ii) holds. Since (i) does not hold, there exists a nonzero vector such that for some . If , then , for all by Remark 2.4. So , for all and hence , which is a contradiction. Then for some nonzero , and hence . Therefore, and .
The following examples show that Proposition 2.7 does not hold for or .
For any positive integer , the linear map defined by , preserves .
The linear map defined by , where , preserves rgw-majorization.
We use the following statements to find the structure of linear preservers of gs-majorization.
Lemma 2.11 (see [6, ]).
Let x and y be two distinct vectors in . Then if and only if and .
If a linear map preserves , then .
Let . For every , it is clear that by Lemma 2.11. Then and hence there exists such that . So and therefore .
Let be a linear map. Then preserves if and only if one of the following holds:
(i)there exists some such that , ,
(ii) for some and ,
Let . Assume that preserves . So by Lemma 2.12. Now, we consider two cases.
Case 1 .
Suppose there exists such that for some . If , then . So and hence . For every , by Lemma 2.11. Then and hence , for all . Then , for some and hence for all . If , consider the basis for . For every , , by Lemma 2.11. Consequently for every and hence for all , where . Therefore, (i) holds in this case.
Case 2 .
Assume that implies . Since , we have for every . Thus it follows that for every , where is the column of and hence . If , then there exists such that and hence . By the hypothesis of this case, . Then (ii) holds. If it is clear (iii) holds.
Conversly, if (i) or (iii) holds it is easy to show that preserves gs-majorization. Suppose that (ii) holds. Then there exists such that . Assume that . If then by Lemma 2.11. If , then there exists such that and hence . Therefore, , and hence . Then and hence preserves gs-majorization.
If preserves and then .
If is injective it is clear that . Assume that is not injective, so there exists a nonzero vector such that . If , then by Case 1 in the proof of Theorem 2.13, for some . Therefore, , which is a contradiction. So and hence . It is clear that , from which and the rank and nullity theorem, we obtain , completing the proof.
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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