A New Determinant Inequality of Positive Semi-Definite Matrices

A new determinant inequality of positive semidefinite matrices is discovered and proved by us. This new inequality is useful for attacking and solving a variety of optimization problems arising from the design of wireless communication systems. I. A NEW DETERMINANT INEQUALITY The following notations are used throughout this article. The notations [·] and [·] stand for transpose and Hermitian transpose, respectively. tr(A) and det(A) denote the trace and the determinant of the matrix A, respectively. The symbols R and R stand for the set of n×m matrices and the set of n-dimensional column vectors with real entries, respectively. C and C denote the set of n × m matrices and the set of n-dimensional column vectors with complex entries, respectively. We introduce the following new determinant inequality. Theorem 1: Suppose A ∈ C and B ∈ C are positive semi-definite matrices with eigenvalues {λk(A)} and {λk(B)} arranged in descending order, D ∈ R is a diagonal matrix with non-negative diagonal elements {dk} arranged in descending order. Then the following determinant inequality holds det ( DAD +B )