FOR A, B Î MNM, WE SAY A IS LEFT MATRIX MAJORIZED BY B AND WRITE (FORMULA) FOR SOME ROW STOCHASTIC MATRIX R. A LINEAR OPERATOR T: MNM® MNM IS SAID TO BE A LINEAR PRESERVER OF (FORMULA) IMPLIES THAT TA<L TB: LINEAR PRESERVERS OF -ℓ ARE CHARACTERIZED. HERE, WE EXTEND THE RESULT TO THE CASE T IS A LINEAR OPERATOR FROM MNM TO MNQ, WHERE M AND Q ARE NOT NECESSARILY IDENTICAL.

LET MN, M BE THE ALGEBRA OF ALL N×M REAL MATRICES. AN N×N REAL MATRIX A IS CALLED CIRCULANT IF THERE EXISTS A ÎRN SUCH THAT (FORMULA) WHERE P IS THE PERMUTATION MATRIX CORRESPONDING TO THE N-CYCLE (12… N). THE SET OF ALL N×N CIRCULANT PERMUTATION MATRICES IS A SUBGROUP OF SN OF ORDER N. IN THIS NOTE, FIRST WE CONSIDER THE CIRCULANT PERMUTATION MATRICES TO INTRODUCE A NEW KIND OF MAJORIZATION AND THEN WE INVESTIGATE SOME PROPERTIES OF THIS CONCEPT.

LET D1 AND D2 BE TWO EUCLIDEAN DISTANCE MATRICES (EDM) WITH CORRESPONDING POSITIVE SEMI DEFINITE MATRICES B1 AND B2 RESPECTIVELY. SUPPOSE THAT L(A) IS THE VECTOR OF EIGENVALUES OF A MATRIX A. IN THIS NOTE WE INVESTIGATE THE CONDITIONS UNDER WHICH L (D1) < L (D2) WHENEVER L (B1) < L (B2). ALSO FOR N = 2 WE SHOW THAT L (D1) <L L (D2) IF AND ONLY IF L (B1) <L L (B2).

LET D1 AND D2 BE TWO EUCLIDEAN DISTANCE MATRICES (EDMS) WITH CORRESPONDING POSITIVE SEMIDEFINITE MATRICESB 1 AND B2 RESPECTIVELY. SUPPOSE THAT (FORMULA) IS THE VECTOR OF EIGENVALUES OF A MATRIX A SUCH THAT (FORMULA). IN THIS PAPER, THE RELATION BETWEEN THE EIGENVALUES OF EDMS AND THOSE OF THE CORRESPONDING POSITIVE SEMIDE FINITE MATRICES RESPECT TO (FORMULA) AND WITH A SPECIAL CONDITION ON R3 WILL BE INVESTIGATED.

LET MN,M BE THE SET OF ALL N×M MATRICES WITH ENTRIES IN F, WHERE F IS THE FIELD OF REAL OR COMPLEX NUMBERS. A REAL OR COMPLEXN N MATRIX IS GENERALIZED ROW STOCHASTIC (G-ROW STOCHASTIC) IF ALL OF ITS ROW SUMS EQUAL ONE. FOR X, Y Î MN,M, WE SAY THAT Y IS LGW-MAJORIZED BY X IF THERE EXISTS AN N×N G-ROW STOCHASTIX MATRIX R SUCH THAT Y=RX. IN THIS PAPER, THE RELATION LGW-MAJORIZATION ON MN,M IS STUDIED, AND THE POSSIBLE STRUCTURE OF ITS LINEAR PRE-SERVERS WILL BE FOUND. ALSO ALL LINEAR OPERATORS T: MN,M®M N,M PRESERVING (OR STRONGLY PRESERVING) LGW-COLUMN MAJORIZATION WILL BE CHARACTERIZED.

AN NONNEGATIVE REAL MATRIX R IS SAID TO BE ROW STOCHASTIC IF ALL ITS ROW SUMS IS EQUAL TO ONE. WE STUDY THE CONCEPT ROW STOCHASTIC: FOR TWO REAL VECTOR X AND Y WE SAY THAT X IS RS-MAJORIZED BY Y (DENOTED BY X <RS Y) IF THERE EXISTS AN N -BY- N ROW STOCHASTIC MATRIX R WITH ALL ITS COLUMN ENTRIES EQUAL SUCH THAT X=RY. THIS PAPER CHARACTERIZES ALL (RESP. STRONG) LINEAR PRESERVERS OF <RS ON R2.