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On Linear Maps Preserving g-Majorization from
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Journal of Inequalities and Applications volume 2010, Article number: 254819 (2010)
Abstract
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.
1. Introduction
Majorization is a topic of much interest in various areas of mathematics and statistics. If and
are
-vectors of real numbers such that
for some doubly stochastic matrix
, then we say that
is (vector) majorized by
; see [1]. Marshall and Olkin
s text [2] is 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]. Let
and
be two vector spaces over a field
, and let
be a relation on both
and
. We say that a linear map
, preserves the relation
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F254819/MediaObjects/13660_2009_Article_2101_Equ1_HTML.gif)
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.
Definition 1.1.
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
.
2. Main Results
In this section we state some preliminary lemmas to describe the linear maps preserving from
to
and the linear maps preserving
or
from
to
.
Lemma 2.1.
Let be a linear map. Then
preserves the subspace
if and only if
.
Proof.
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
.
Theorem 2.3.
A nonzero linear map preserves
if and only if
and
.
Proof.
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
.
Lemma 2.5.
Let be a linear map. If
implies
, then T preserves
.
Proof.
Let and
. If
then
and it is clear that
. If
so
by the hypothesis and hence
, by Remark 2.4. Therefore
preserves
.
Lemma 2.6.
Let be a nonzero singular linear map. Then
preserves
if and only if
and
.
Proof.
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.
Proposition 2.7.
Let be a nonzero linear preserver of
. Then
.
Proof.
If is injective, then
. If
is not injective, we obtain
by Lemma 2.6 and
. Therefore
, by the rank and nullity theorem.
Theorem 2.8.
Let be a nonzero linear map and
. Then
preserves
if and only if one of the following holds:
(i),
(ii) and
.
Proof.
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
.
Example 2.9.
For any positive integer , the linear map
defined by
, preserves
.
Example 2.10.
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
.
Lemma 2.12.
If a linear map preserves
, then
.
Proof.
Let . For every
, it is clear that
by Lemma 2.11. Then
and hence there exists
such that
. So
and therefore
.
Theorem 2.13.
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
,
(iii) and
Proof.
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.
Corollary 2.14.
If preserves
and
then
.
Proof.
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.
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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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Armandnejad, A., Heydari, H. On Linear Maps Preserving g-Majorization from to
.
J Inequal Appl 2010, 254819 (2010). https://doi.org/10.1155/2010/254819
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DOI: https://doi.org/10.1155/2010/254819