**KMS º½ÇÐÈ¸ Æ÷Ç×°ø´ë 2000. 4. 22. (Åä) **** **

On Linear Preservers of Several

Multivariate Majorization

** Contents :**

** **** (0) Abstract**

** (1) Definitions**

** (2) History **

** (3) Previous Results**

** (4) Main Results**

** **** **** ¼º±Õ°ü´ëÇÐ±³
¼öÇÐ°ú **

** ÀÌ
»ó ±¸ **

** **** KMS º½ÇÐÈ¸ Æ÷Ç×°ø´ë 2000.
4. 22. (Åä) **** **

** **

** (0) Abstract**

** **

**In
this paper, ****we
characterize linear operators on real matrices that preserve multivariate majorization.**** We place ****no additional restrictions ****on the linear operator.**

** **

** (1) Definitions**

** **

**Majorization
is a topic of ****much
interest in various areas of mathematics and statistics. (1929³â Hardy, Littlewood.
Polya's : convex ft. g °¡ ÀÏ ÇÊ¿äÃæºÐÁ¶°Ç) (1923³â Schur's work on Hadamard ºÎµî½Ä)(1905³â °æÁ¦ÇÐ¿¡¼
Lorenz curve)- ´ë¼ö, ÇØ¼®, ±âÇÏ, ¼öÄ¡ÇØ¼®, Åë°è, °æÁ¦µî **

** **

**If and are nonincreasing -vectors of nonnegative real numbers such that for with equality at , then we say that is ***(vector)
majorized***
by and write. ****Equivalently,**** if two row vectors have the relationship
then there is a doubly stochastic matrix such that . ****(ÀÌ ¿Ü¿¡
¿¬¼ÓÇÔ¼ö version, ¹«ÇÑ ¼ö¿ version µîµµ ÀÖÀ½)**

** **

**Our interest is in the subject of ****majorization for matrices****. The ****MATRIX VERSION****
****of majorization**** is that**** an real matrix is multivariate majorized by **** ****if there is a ****DOUBLY STOCHASTIC matrix**** such that . **** **

**G. Dahl****'s
Definition in Matrix majorization, Linear Algebra Appl. 288:53-73(1999)****--- "Row Stochastic Matrix"**

** **

**A linear operator is said to ****preserve
a relation ****R ****on ****if ****R**** **

**implies ****R****.**

** **

*** The topic of linear preservers (ºÎ·Ï) **

** **** **

** (2) History **

** **

**In
****"Majorization
and inequalities in matrix theory"****, Linear Algebra Appl., V.199:17-67 (1994), ****T. Ando characterized the linear
operators which preserve vector majorization ****(equivalently, that preserve multivariate majorization
on ****. We
shall characterize all linear operators which preserve multivariate majorization
with no conditions on m, n, or T.**

** **

***. "Linear Operators ****Strongly Preserving ****Multivariate Majorization " (with Y.H. Lee) Proc.
Workshops in Pure Math. V. 17, part I (1997) pp.113-118. **

** **

****. Linear operators ****strongly preserving**** multivariate majorization**** with $T(I)=I$****. Kyungpook Math. J. 39 (1999), no. 1, 191--194. (with
L. Beasley and Y.-H. Lee) (Reviewer: Chi-Kwong Li) 15A04 (15A21 15A30)**

** **

*****. ****A
proof of conjecture****
on strong preservers of multivariate majorization" (with L. Beasley), preprint.**

** **

**C.-K. Li, B.-S. Tam. N.-K. Tsing, ****Linear maps preserving the set of doubly stochastic
matrices, ****Preprint(1999)'s
Main Result : **** TFAE**

**(a) T(DS(n)) = DS(n)**

**(b) T(P(n)) = P(n)**

**(c) TÀÇ ¸ð¾çÀº for some P, Q in P(n)**

**µû¶ó¼ **

**then So let ·Î ³õÀ¸¸é Á¶°Ç T(I) =I¸¦ ¾ø¾Ù ¼ö ÀÖ´Ù°í »ý°¢ÇßÀ½.**

****** Linear Operators ****Preserving****
Multivariate Majorization " (with L. Beasley), Linear Algebra and its Applications
304 141-159 (2000) (
TODAY!! )**

** **

** Linear Operators Strongly Preserving ****Matrix Majorization**** " (with Y.-H. Lee and L.
Beasley), Preprint**

** **

** (3) Previous Results**

**NOTE: ¾Æ·¡¿¡¼ ?? Àº ½Ç¼ö R À» ÀÇ¹Ì ÇÑ ´ä´Ï´Ù.
Áï, ****Àº ****ÀÔ´Ï´Ù****. **

(editor ÀÇ error)

**(´ë¿ì¼ø¼ö¼öÇÐ 1997) Theorem 2.11 **** Let for . Then is a multivariate majorization strong preserver on if one of the following or a composition of the following holds :**

** (1) for any nonzero scalar , an invertible matrix and a permutation matrix ;**

** (2) for a linear functional .**

** **

**(°æºÏ¼öÇÐÀú³Î 1999) Theorem 3.1 ****If strongly preserves multivariate majorization and (i.e., preserves nonnegative matrices), then a permutation matrix .**

** **

** (4) Main Results**

** **

** By
Birkhoff's Theorem , DS(n) is the convex hull of the set P(n) of n by n permutation
matrices. **

** We
now ****define two
linear mappings****,
which are fixed throughout the paper.**

** **

**Definition
2.1 **

**Define
by where**

**. That is,**** is the ***column*** vector whose th component is the th row sum of , ****equivalently
. Define by where ****Thus, is the diagonal matrix whose main diagonal is the vector .**

** **

**Lemma 2.2**

**If is any element of and is any element of , then and **

** **** D¿Í
rÀÌ doubly stochastic Çà·Ä °öÇÑ °Í¿¡ ºÒº¯ **

**Proof. **

**Let and . Let . Then the entry of is **

** **

**That is, . Similarly .**

** **

**Lemma 2.4**

**Let and the linear operator be defined by , then preserves multivariate majorization.**

** **

**MAIN THEOREM 2.5 **

**Let be a linear operator on . Then TFAE:**

**(i) preserves multivariate majorization.**

**(ii) Either**

** (a) There exist matrices and such that for all or**

** (b) There exist matrices and a permutation matrix such that for all . **

** **

**The remaining part of this article is devoted to the proof
of the Main Theorem. **

** **

**We state here ****the row version of Ando's characterization.**

** **

**Lemma 2.6 (Ando, Corollary 2.7 ) Let preserve (vector) majorization, then either**

**(i) for some or**

**(ii) for some and**

** .**

** **

**Lemma 2.7**

**Let the linear operator on preserve multivariate majorization and . ****Then**

**preserves
multivariate majorization.**

** Áõ¸í
: T´Â M.M.P. ÀÎµ¥ S°¡ ¾Æ´Ï¶ó¸é ¸ð¼øÀÓ. **

** **

** **** ¿©±â¼ ¿ì¼± T(X)ÀÇ j-th
row¸¦ »ý°¢ÇÑ´Ù.**

** **

**Lemma
2.8 **

**If
preserves multivariate majorization, ****then defined by **

**the th row of T [ 0 ... 0 x 0 ... 0 ]^t**

** i^th
row**

** preserves (vector) majorization.**

** **

** ¿©±â¼ º¸ÀÌ´Â °ÍÀº
°¡ m.m.p.ÀÌ¹Ç·Î ´Â °¢°¢ÀÇ row vector maj.À» º¸Á¸ÇÑ´Ù´Â °Í. ±×·¯¸é ÀÌ¾î¼ T°¡ linear ÀÌ¹Ç·Î
´ÙÀ½À» °®´Â´Ù.**

** **

**Lemma 2.9**

**If preserves multivariate majorization, then for and ****there exist
****vectors , permutation matrices , and scalars and **

** with or , and , such that**

** **

** **

** ****ÀÌÁ¦
ÀÌ imageÀÇ ¸ð¾çÀ» ´Ü¼øÈ÷ ÇÏ±â À§ÇØ °è¼öÀÇ ¼ºÁúµéÀÇ °¡´É¼ºÀ» ÁÙ¿©°£´Ù.**

** **

**Lemma 2.10 **

**Suppose for some is not a multiple of , we have **

**If preserves multivariate majorization ****then necessarily for all .**

** **

**Lemma 2.12 If preserves multivariate majorization and**

** **

**has at least two nonzero rows, one of which, say the th, is for some nonzero vector which is not a multiple of, and the other, say the th, is for some and ,**** then .**

** **

**ÀÌÁ¦ µÎ °³ÀÇ ¸ð¾çÀÌ µ¿½Ã¿¡ ³ªÅ¸³ª¸é ¸ð¼øÀÓÀ» º¸ÀÎ´Ù.**

** **

**Lemma 2.13**

**If **

**for some which is not a multiple of , then does not preserve multivariate majorization.**

** **

** **** ´ÙÀ½ÀÇ
µÎ Lemma´Â and ÀÓÀ» º¸¿©ÁØ´Ù. **

** **

**Lemma 2.15**

**Let be defined by **

**where . ****If preserves multivariate majorization, then .**

** **

**Lemma 2.16**

**Let be defined by **

**where . ****If preserves multivariate majorization, then .**

** **

**Now we can prove Main Theorem.**

** **

**MAIN THEOREM 2.5 **

**Let be a linear operator on . Then TFAE:**

**(i) preserves multivariate majorization.**

**(ii) Either**

** (a) There exist matrices and such that for all or**

** (b) There exist matrices and a permutation matrix such that for all . **

**PROOF**

**From the Corollary (2.14), T(X) must have one of the forms:**

**Either**

**(i) There exist matrices and such that for all or**

**(ii) T(X) =[] + for some , in , and for all in .**

** In the second case, from the previous Lemmas 2.15, 2.16
and 2.7 we have **

**T(X) =[] + = for some matrix and permutation matrix . ****QED.**

** **

** **

**(2) ****Linear
Preserver Problems **

** **

** Let K be an algebraic structure and let M(K) denote
the vector space of all m*n matrices over K. **

** Over the last century, a great
deal of effort has been devoted to the following problem. ****Characterize those linear
operators T : M --> M which leave a function or set
invariant.****
We call this a ****linear preserver problem****. **

** The most typical and oldest
type of linear preserver problem is as follows:**

** **

**(1)ÀÏ ¶§ ¾Æ·¡ ´Â?**

** **

** ****F
´Â scalar valued, matrix valued or set-valued function on M****. **

** **

** It is clear that the set
of all such matrix-valued linear operators ****T is a multiplicative semigroup
with an identity****.
**

** The study of these operators began in ****1897**** when ****Frobenius**** characterized the ****linear operators that preserve the determinant over two
different sets of matrices ****:**

** Frobenius proved that the semigroup
of linear operators that preserve the determinant consists
of linear transformations of the form **

** (1.1)**

** **

** (i) C °¡ complex field ÀÎ °æ¿ì, det
(UV) = 1**

** (ii) M ÀÌ real symmetric matrices ÀÎ
°æ¿ì, **

** U
= kQ, V = Q^T and det(Q).**

** **

** In 1925****, ****Schur
extended and improved ****the
result for Case (i) as follows. Let ?(A) be the (nCk)^2 -
tuple of the k-th order (i.e., kxk) minors of A in some order where
k ? 3 is fixed. **

** Schur proved that if ?(T(A)) = S(?(A)),
for a fixed nonsingular matrix S, then T has one of the two forms in (1.1) (without
the restriction det (UV) = 1). **

** **

** In 1949****, Dieudonné showed ****that if F is any field and if T
is a semilinear operator mapping M(F) onto itself, which holds
the cone det (A) = 0 invariant, then T is of the form **

** **

**where ? is an automorphism of the field and a_ij is the (i,j)
entry of A.**

** We will call those operators T in (1.1)
**

** (U,V)-operators****. **

** **

**Other interesting results**** of this type include:**

** **

** ****(a)
[59] If M is is the set of all nxn Hermitian matrices, **

** and F(A) = Inertia (A), then
T is a (U,V)-operator**

** where U = V*, the conjugate
transpose of U, for **

** some nonsingular matrix V ?
M(C).**

** **

**(b) [43] If M = M (C) and F(A) = { x*Ax| x ? C , x*x
= 1}, **

** then T is a (U,V)-operator
where U = V* for some **

** unitary matrix U.**

** **

** Consequences and recent work
of this type can be found in [3, 17,
22, 23, 24, 28, 48, 53, 54] for preservers of generalized
matrix functions, in [33, 39, 40, 50, 55] for inertia preservers, and
in [44, 45, 46, 47] for numerical range and radius preservers.**

** **

** We note that ****many linear preserver problems
can be reduced****
to the problem of characterizing linear operators on M(K) which map the set
of rank 1 matrices into itself.**

** **

** A ****semiring ****(Gregory and Pullman [32] or Kim [41]) is a
binary system (S, +, ?) such that (S, +) is an Abelian monoid (identity
0), (S, ?) is a monoid (identity 1), ? distributes over +, 0 ? s = s ?
0 = 0 for all s in S, and 1 $ 0. If 0 is the only element
to have an additive inverse, then S is antinegative.**

** **

**The Second type of linear preserver problem is:**

**(2) Let ÀÏ ¶§ ´Â s.t**

** **

** ****Some examples **** of this type of problem with
**

**appropriate characterizations.**

** **

** (a) [51] If M =
M(C), ****B is the
set of unitary**

** matrices in M(C) and T(B) ¡ø
B****, then T is a**

** ****(U,V)-operator for some unitary matrices U and V****.**

** **

** (b) [4, 11, 13, 53] Suppose M = M(C)
and ****B is the**

** set of all rank-k matrices****. If T(B)¡øB, then T is**

** a ****(U,V)-operator for some nonsingular matrices**

** U and V****.**

** **

** (c) [19] Suppose M = M(S) where
****S is an **

** antinegative semiring with no zero
divisors, and B**

** is the set of all idempotent matrices**** in M(S). If**

** T(B) = B, then T is a ****(U,V)-operator where **

** U = V for some invertible
matrix V in M(S).**

** **

** We may note that,
in the first example, we were simply characterizing
the structure of all linear operators T that map the unitary
group into itself.**

** **

** ****There is an extensive literature concerning the characterization
of linear operators of type [II]****. Further results may be found in [1, 2, 6, 7, 8, 9,
30, 49] for sets defined by rank, in [15, 16, 18, 21, 29] for
sets defined by graph theoretic
properties, i.e., irreducibility or strong connectivity,
term rank, etc., and in [10, 20, 25, 35, 56, 57] for characterization
of preservers of other sets.**

** **

** The last and 3rd type of linear preserver problem
****is: (3) ÀÎ ´Â?**

** Some
of the major results are:**

** (a) [5, 61] Suppose
M = M (C) and A ~ D if A**

** and D commute. If T is
a nonsingular linear**

** operator on M satisfying T(A) ~ T(D)
whenever **

** A ~ D, then T is of the form **

** or **

** **** **

** for some nonsingular
matrix X ? M (C) ; l ? C**

** and linear functional
f on M.**

** **

** (b) [62] Suppose M = M (C) and
A ~ D if A and**

** D are similar. **

** If T
is a linear operator on M satisfying **

** T(A)~T(D) whenever A~D, then T is
of the form**

** **** **

** or **

** **** **

** for some nonsingular matrix X ? M
(C), l ? C and **

** linear functional f on
M.**

** **

**Further studies of the above type are found in [36, 61].**

** In [26] and [27], Chan,
Lim and Tan gave the following characterization
for those linear operators that preserve idempotent and tripotent
matrices. **

** THE
END **

** ************************

**Âü°í
ÀÚ·á:**

**A.
W. Marshall and I. Olkin, ****Inequalities:
Theory of Majorization and Its Applications,**** Academic Press, 1979**

** **

**T.
Ando,**** Majorization,
doubly stochastic matrices and comparison of eigenvalues,**** Linear Algebra Appl. V.118:163-248 (1989).**

** **

**T.
Ando,**** Majorization
and inequalities in matrix theory,**** Linear Algebra Appl., V.199:17-67 (1994)**

** **

**J.
V. Bondar, ****Comments
and complements to: Inequalities: Theory of Majorization and Its Appl. by Albert
W. Marshall and Ingram Olkin,****
Linear**

**Algebra
Appl. V.199:115-130(1994)**

** **

**G.
Dahl, ****Matrix majorization,**** Linear Algebra Appl. 288:53-73(1999)**

** **

** *******************************