On a question of Bhatia and Kittaneh
Copyright policy )On a question of Bhatia and Kittaneh
This paper settles in the affirmative a question of Bhatia and Kittaneh. For positive semidefinite \(n\times n\) matrices \(P\) and \(Q\), \(\sqrt {\sigma_r(PQ)} \leq \frac 12\lambda_r(P+Q)\) holds for \(r=1, \dots, n\) where \(\sigma_k(M)\) denotes the \(k\)th largest singular values of \(M\) and \(\lambda_k(H)\) denotes the \(k\)th largest eigenvalue of a Hermitan matrix. The consequence \(\|| \,|AB|^{1/2}\, \|| \leq \frac 12 \|| A+B\||\) is true for all unitarily invariant norm \(\||\cdot\||\). The proof uses the Schur complement.
largest singular values, Numerical Analysis, Arithmetic mean, Algebra and Number Theory, unitarily invariant norm, Eigenvalue, Hermitan matrices, Inequalities involving eigenvalues and eigenvectors, Geometric mean, arithmetic mean, Singular value, geometric mean, Positive definite matrix, positive definite matrices, Norms of matrices, numerical range, applications of functional analysis to matrix theory, Discrete Mathematics and Combinatorics, Hermitian, skew-Hermitian, and related matrices, Schur complement, Geometry and Topology, largest eigenvalue
largest singular values, Numerical Analysis, Arithmetic mean, Algebra and Number Theory, unitarily invariant norm, Eigenvalue, Hermitan matrices, Inequalities involving eigenvalues and eigenvectors, Geometric mean, arithmetic mean, Singular value, geometric mean, Positive definite matrix, positive definite matrices, Norms of matrices, numerical range, applications of functional analysis to matrix theory, Discrete Mathematics and Combinatorics, Hermitian, skew-Hermitian, and related matrices, Schur complement, Geometry and Topology, largest eigenvalue
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