Let Mm, nMm, n be the set of all mm-by-nn real matrices. A matrix RR in Mm, nMm, n with nonnegative entries is called strictly sub row stochastic if the sum of entries on every row of RR is less than 1. For A, B∈ Mm, nA, B∈ Mm, n, we say that AA is strictly sub row Hadamard majorized by BB (denoted by A≺ SHB)A≺ SHB) if there exists an mm-by-nn strictly sub row stochastic matrix RR such that A=R∘ BA=R∘ B where X∘ YX∘ Y is the Hadamard product (entrywise product) of matrices X, Y∈ Mm, nX, Y∈ Mm, n. In this paper, we introduce the concept of strictly sub row Hadamard Majorization as a relation on Mm, nMm, n. Also, we find the structure of all linear operators T: Mm, n→ Mm, nT: Mm, n→ Mm, n which are preservers (resp. strong preservers) of strictly sub row Hadamard Majorization.