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UM E-Theses Collection (澳門大學電子學位論文庫)

Title

Equality cases for some inequalities involving the Hadamard product of Hermitian matrices

English Abstract

In this thesis, we study the equality cases for some inequalities involving the Hadamard product of Hermitian matrices. Let Cₙₓₙ denote the set of all n x n complex matrices, and Hₙ be the set of all n x n Hermitian matrices. The Hadamard (entrywise) product of A=(aᵢⱼ), B(bᵢⱼ) ∈ Cₙₓₙ is defined and denoted by A ◦ B=(aᵢⱼbᵢⱼ) ∈ Cₙₓₙ. For any A ∈ Hₙ (resp. A ∈ Cₙₓₙ), let λ₁(A) ≥ ... ≥ λₙ(A) (resp. σ₁(A) ≥ ... ≥ σₙ(A)) denote the eigenvalues (resp. singular values) of A and let λ(A) = (λ₁(A), ..., λₙ(A))ᵗ (resp. σ(A) =(σ₁(A), ..., σₙ(A))ᵗ). For any n x n positive definite matrices A and B, it is known that the following inequalities in multiplicative form hold: ∏(i = k, n)λᵢ(A ◦ B) ≥ ∏(i = k, n)λᵢ(AB), 1 ≤ k ≤ n, ∏(i = k, n)λᵢ(A ◦ B) ≥ ∏(i = k, n)λᵢ(ABᵗ), 1 ≤ k ≤ n. In Chapter 2, we characterize the equality cases of the above inequalities. For general matrices A, B ∈ Cₙₓₙ, it is known that ∑(i = 1, k)σᵢ(A ◦ B) ≤ ∑(i = 1, k)σᵢ(A)σᵢ(B), 1 ≤ k ≤ n. Moreover, the equality cases are also known. However, when A and B are restricted to be Hermitian, further consideration is needed. In Chapter 3, we continue the study and characterize the equality of the above inequalities for Hermitian matrices. A brief introduction of the inequalities is given in Chapter 1.

Issue date

2001.

Author

Law, Ieng Chi

Faculty

Faculty of Science and Technology

Department

Department of Mathematics

Degree

M.Sc.

Subject

Matrices

Matrix inequalities

Supervisor

Cheng, Che Man

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Location
1/F Zone C
Library URL
991008432739706306