UM E-Theses Collection (澳門大學電子學位論文庫)
- Title
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Equality cases for some inequalities involving the Hadamard product of Hermitian matrices
- English Abstract
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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
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2001.
- Author
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Law, Ieng Chi
- Faculty
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Faculty of Science and Technology
- Department
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Department of Mathematics
- Degree
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M.Sc.
- Subject
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Matrices
Matrix inequalities
- Supervisor
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Cheng, Che Man
- Files In This Item
- Location
- 1/F Zone C
- Library URL
- 991008432739706306