On some matrix operator and its applications

The paper focuses on a matrix operator, which maps a square real matrix to a block matrix (called the saddle point matrix), where the left-up block represents the given matrix, the right-down block is zero, and two other blocks are vectors of ones. The operator transforms any symmetric matrix into the Karush-Kuhn-Tucker matrix of standard quadratic program on the standard simplex, which is the intersection of a hyperplane with the positive orthant. There are shown some properties of this matrix operator, connections with game theory and necessary and sufficient conditions for existence of unique interior optimizer of standard quadratic program.