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 .

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)

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