نگهدارنده‌هاي خطي احاطه سازي زوج روي Mn,mو نگهدارنده‌هاي خطي احاطه سازي دوار روي Rn .

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دكتري

Let AϵM_(n,m) be the algebra of all n×m real matrices. An n×n real matrix A is even doubly stochastic if it is a convex combination of even permutation matrices. For A,BϵM_(n,m), A is said to be even majorized by B (written as A≺_e B), if there exists an even doubly stochastic matrix D such that A=DB. Also a circulant doubly stochastic matrix B is a convex combination of circulant permutation matrices. For x,y∈R^n, x is said to be circulant majorized by y (written as x≺_c y), if there exists a circulant doubly stochastic matrix D such that x=Dy. Let R be a relation on R^n. A linear operator T:R^n→R^n is said to be a linear preserver of R if for all x,y∈R^n

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