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.