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For two <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow></semantics></math></inline-formula> real matrices <i>X</i> and <i>Y</i>, <i>X</i> is said to be majorized by <i>Y</i>, written as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>≺</mo><mi>Y</mi></mrow></semantics></math></inline-formula> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>=</mo><mi>S</mi><mi>Y</mi></mrow></semantics></math></inline-formula> for some doubly stochastic matrix of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>.</mo></mrow></semantics></math></inline-formula> Matrix majorization has several applications in statistics, wireless communications and other fields of science and engineering. Hwang and Park obtained the necessary and sufficient conditions for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow></semantics></math></inline-formula> to satisfy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>≺</mo><mi>Y</mi></mrow></semantics></math></inline-formula> for the cases where the rank of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Y</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> and the rank of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Y</mi><mo>=</mo><mi>n</mi></mrow></semantics></math></inline-formula>. In this paper, we obtain some necessary and sufficient conditions for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow></semantics></math></inline-formula> to satisfy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>≺</mo><mi>Y</mi></mrow></semantics></math></inline-formula> for the cases where the rank of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Y</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>2</mn></mrow></semantics></math></inline-formula> and in general for rank of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Y</mi><mo>=</mo><mi>n</mi><mo>−</mo><mi>k</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>. We obtain some necessary and sufficient conditions for <i>X</i> to be majorized by <i>Y</i> with some conditions on <i>X</i> and <i>Y</i>. The matrix <i>X</i> is said to be doubly stochastic majorized by <i>Y</i> if there is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>∈</mo><msub><mi mathvariant="sans-serif">Ω</mi><mi>m</mi></msub></mrow></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>=</mo><mi>Y</mi><mi>S</mi></mrow></semantics></math></inline-formula>. In this paper, we obtain some necessary and sufficient conditions for <i>X</i> to be doubly stochastic majorized by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Y</mi><mo>.</mo></mrow></semantics></math></inline-formula> We introduced a new concept of column stochastic majorization in this paper. A matrix <i>X</i> is said to be column stochastic majorized by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Y</mi><mo>,</mo></mrow></semantics></math></inline-formula> denoted as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><msup><mo>⪯</mo><mi>c</mi></msup><mi>Y</mi><mo>,</mo></mrow></semantics></math></inline-formula> if there exists a column stochastic matrix <i>S</i> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>=</mo><mi>S</mi><mi>Y</mi><mo>.</mo></mrow></semantics></math></inline-formula> We give characterizations of column stochastic majorization and doubly stochastic majorization for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> matrices.