关于矩阵行列式与迹的一个不等式

Some Inequalities for the Determinant and Trace of the Positive Semidefinite Hermitian Matrices

本文证明了ｎ阶半正定Ｈｅｒｍｉｔｉａｎ矩阵Ａ≥Ｂ的特征值不等式∑Ｋｔ＝１ａｒｔ（Ｂ）∑Ｋｔ＝１ａｒｔ（Ａ）≥∏Ｋｔ＝１λｒｔ（Ｂ）∏Ｋｔ＝１λｒｔ（Ａ）

Let λ 1(A)≥…≥λ n(A) denote the eigenvalues of a Hermitian by matrix A of n order,Our main results are∑Kt=1λ r t(B)∑Kt=1λ r t(A)≥∏Kt=1λ r t(B)∏Kt=1λ r t(A),K=1,2,…,S=rk(A)Here A,B are positive semidefinite Hermitian matrices of n order and A≥B.