Matrix Inequalities by Means of Block Matrices 1

One of the most useful tools for deriving matrix inequalities is to utilize block matrices; usually they are 2× 2 in most applications. In this paper, we shall show a weak log-majorization inequality of singular values for partitioned positive semidefinite matrices, from which some classical and recent results of Bhatia and Kittaneh [4], Wang, Xi and Zhang [12], and Zhan [13] will follow. We shall also develop a new technique that is complementary to the Schur complement; while by making use of Schur complements, a number of determinantal, trace, and other inequalities are exhibited in [16]. With the new technique we add more inequalities to these in [16]. We denote the eigenvalues of an n×n complex matrix X by λi(X), i = 1, 2, . . . , n, and arrange them in modulus decreasing order |λ1(X)| ≥ |λ2(X)| ≥ · · · ≥ |λn(X)|. The singular values of an m × n matrix X are denoted by σ1(X), . . . , σn(X) and are also arranged in decreasing order. Note that σi(X) = λi(|X |) for each i, where |X | = (X∗X) 2 . We further write