Bo-Ying Wang

Some inequalities for the eigenvalues of the product of positive semidefinite Hermitian matrices

Let λ1(A)⩾⋯⩾λn(A) denote the eigenvalues of a Hermitian n by n matrix A, and let 1⩽i1< ⋯ <ik⩽n. Our main results are ∑t=1kλt(GH)⩽∑t=1kλit(G)λn−it+1(H) and ∑t=1kλit(GH)⩽∑t=1kλit(G)λn−t+1(H) . Here G and H are n by n positive semidefinite Hermitian matrices. These results extend Marshall and Olkins inequality ∑t=1kλt(GH)⩽∑t=1kλt(G)λn−t+1(H) . We also present analogous results for singular values.

Schur complements and matrix inequalities of hadamard products

Let A and B be n-square positive definite matrices. Denote the Hadamard product of A and B by A∘ B. The main results of the paper are: 1.For any matrices C and D of size m × n and 2.Let A/α be the Schur complement of A(α) in A. Then . Some other matrix inequalities of Schur complements and Hadamard products of positive definite matrices are also presented.

Some inequalities on generalized Schur complements

Abstract This paper presents some inequalities on generalized Schur complements. Let A be an n×n (Hermitian) positive semidefinite matrix. Denote by A/α the generalized Schur complement of a principal submatrix indexed by a set α in A. Let A + be the Moore–Penrose inverse of A and λ(A) be the eigenvalue vector of A. The main results of this paper are: 1. λ(A + (α′))⩾λ((A/α) + ) , where α′ is the complement of α in {1,2,…,n} . 2. λ(A r /α)⩽λ r (A/α) for any real number r⩾1. 3. (C * AC)/α⩽C * /α A(α′) C/α for any matrix C of certain properties on partitioning.

A high symmetry class of tensors with an orthogonal basis of decomposable symmetrized tensors

We present a high symmetry class of tensors with an orthogonal basis of decomposable symmetrized tensors, and this is a counter-example of the claim presented in [1].

On the Hadamard product of inverse M-matrices

Abstract We investigate the Hadamard product of inverse M -matrices and present two classes of inverse M -matrices that are closed under the Hadamard multiplication. In the end, we give some inequalities on the Fan product of M -matrices and Schur complements.

Some eigenvalue inequalities for positive semidefinite matrix power products

Abstract We study the eigenvalues of positive semidefinite matrix power products and obtain some inequalities, most of which are in terms of majorization. In particular, for A, B ⩾ 0, β > α > 0, we prove log λ 1 α (AαBα) ≺ log λ 1 β (AβBβ).The result is a generalization of some work of Marcus, Lieb, Thirring, Le Couteur, Bushell, and Trustrum.

A Trace Inequality for Unitary Matrices

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Some inequalities for sum and product of positive semidefinite matrices

Abstract The purpose of this paper is to present some inequalities on majorization, unitarily invariant norm, trace, and eigenvalue for sum and product of positive semidefinite (Hermitian) matrices. Some open questions proposed by Marshall and Olkin are resolved.

Some inequalities for singular values of matrix products

Abstract Let α1(C) ≥ … ≥ αn(C) denote the singular values of a matrix C e C n×m, and let 1 ≤ i1 R . The main results are Σt=1kΣitr(A)Σn−it+1(B), where A e C p×n, B e C n×m. We also consider the cases for the product of three matrices and more.

The subspaces and orthonormal bases of symmetry classes of tensors

We present here the dimensions of some subspaces in the symmetry classes of tensors and some methods for constructing orthonormal bases of the symmetry classes of tensors.