2016/11/16 by Tin-Yau Tam, Tam, Tin-Yau, Pingping Zhang +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Graph theory and applications #Mathematical Inequalities and Applications #Point processes and geometric inequalities #math.FA
paper · pdf · doi:10.48550/arxiv.1611.05108
arxiv created 2016/11/16 · openalex publication_date 2016/11/16 · arxiv updated 2016/11/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Denote by ¶n the set of n× n positive definite matrices. Let D = D1⊕ … ⊕ Dk, where D1∈ ¶n1, …, Dk ∈ ¶nk with n1+⋯ + nk=n. Partition C∈ ¶n according to (n1, …, nk) so that \Diag C = C1⊕ … ⊕ Ck. We prove the following weak log majorization result: λ(C-11D1⊕ ⋯ ⊕ C-1kDk)\precw log λ(C-1D), where λ(A) denotes the vector of eigenvalues of A∈ \Cnn. The inequality does not hold if one replaces the vectors of eigenvalues by the vectors of singular values, i.e., s(C-11D1⊕ ⋯ ⊕ C-1kDk)\precw log s(C-1D) is not true. As an application, we provide a generalization of a determinantal inequality of Matic \cite[Theorem 1.1]M. In addition, we obtain a weak majorization result which is complementary to a determinantal inequality of Choi \cite[Theorem 2]C and give a weak log majorization open question.