In this thesis , we present characterization of Hadamard powers preserving Loewner positivity , monotonicity , and convexity based on an article by Guillot , Khare , and Rajaratnam (2015). Hadamard powers of positive definite matrices that preserving a property with respect to the Loewner ordering arise in various applications . Let e a function and e an matrix , the Hadamard function of is defined by which is obtained by applying to the entries of . We study the properties of entrywise powers of , that is the properties of when is applied elementwise to for . For the even and odd multiplicative extensions to of the power function are defined by and <v:imagedata src="file:///C:\\Users\\hajjar\\AppData\\Local\emp\\msohtmlclip1\\01\\clip_image014.png" chromakey="white" o:tit
