题目：New Rank Inequalities for the Hadamard Product with Applications
摘要：The Hadamard or elementwise product of matrices is fundamental in matrix analysis. Its study dates back to the early 20th century due to the mathematician Jacques Hadamard. The most well-known and also probably most important result is Schur product theorem, stating that the Hadamard product of positive (semi)definite matrices remains to be positive (semi)definite. In this talk, we will revisit the developments of Schur’s theorem in the past century. We provide two new rank inequalities for the Hadamard product, giving sufficient conditions under which the Hadamard product of positive semidefinite matrices is strictly positive definite. Applications of the new results are presented in matrix theory as well as in array processing to resolve a fundamental problem as to how many antennas are required to resolve a fixed number of sources. As a byproduct, we obtain a new rank inequality for the Khatri-Rao product of matrices which should be of independent interest.