Bull. Korean Math. Soc. 39 (2002), No. 2, pp. 283–287 RESOLUTION OF THE CONJECTURE ON STRONG PRESERVERS OF MULTIVARIATE MAJORIZATION LeRoy B. Beasley, Sang-Gu Lee * , and You-Ho Lee ** Abstract. In this paper, we will investigate the set of linear op- erators on real square matrices that strongly preserve multivariate majorization without any additional conditions on the operator. This answers earlier conjecture on nonnegative matrices in [3]. 1. Introduction Majorization is a topic of much interest in various areas of mathemat- ics and statistics. If x and y are nonincreasing n-vectors of nonnegative real numbers such that ∑ k i=1 x i ≤ ∑ k i=1 y i for k =1,...,n with equality at k = n, then we say that x is (vector) majorized by y and write x ≺ y. Our interest is in the subject of majorization for matrices. The matrix version of majorization is that an m × n real matrix A is majorized by B if there is a matrix X such that A = BX . We are interested in a specific type of majorization of matrices. One such possibility is to say that A is (matrix) majorized by B, written A ≺ mul B, if there exists a row stochastic matrix X such that A = BX , which was defined in [8]. Another possibility is to say that A is multivariate majorized by B if there exists a doubly stochastic matrix D such that A = BD. This is motivated by the theorem of Hardy-Littlewood and Polya saying that for row vectors a and b in R n , a is to be majorized by b, written a ≺ b, if there exists an n × n doubly stochastic matrix D such that a = bD. References on multivariate majorization are found in [1, 2, 4, 7, 13]. Let DS(n) be the set of n × n doubly stochastic matrices, i.e., non-negative matrices with all row sums and column sums equal to one. The set Received June 23, 2000. 2000 Mathematics Subject Classification: 15A04, 15A21, 15A30. Key words and phrases: majorization, multivariate majorization, strong preserver, doubly stochastic matrix . This work was partially supported by KRF 2001-DP0005 * and the BK21 program ** .