Inequalities for selected eigenvalues of the product of matrices

The product of a Hermitian matrix and a positive semidefinite matrix has only real eigenvalues. We present bounds for sums of eigenvalues of such a product.Comment: to appear in AMS Proceeding

Positivity of Partitioned Hermitian Matrices with Unitarily Invariant Norms

We give a short proof of a recent result of Drury on the positivity of a $3\times 3$ matrix of the form $(\|R_i^*R_j\|_{\rm tr})_{1 \le i, j \le 3}$ for any rectangular complex (or real) matrices $R_1, R_2, R_3$ so that the multiplication $R_i^*R_j$ is compatible for all $i, j$, where $\|\cdot\|_{\rm tr}$ denotes the trace norm. We then give a complete analysis of the problem when the trace norm is replaced by other unitarily invariant norms.Comment: 6 page

Matrix Inequalities by Means of Block Matrices

We first show a weak log-majorization inequality of singular values for partitioned positive semidefinite matrices which will imply some existing results of anumber ofauthors, then present some basic matrix inequalities and apply them to obtain a number of matrix inequalities involving sum, ordinary product and Hadamard produc

Geršgorin type theorems for quaternionic matrices

AbstractThis paper aims to set an account of the left eigenvalue problems for real quaternionic (finite) matrices. In particular, we will present the Geršgorin type theorems for the left (and right) eigenvalues of square quaternionic matrices. We shall conclude the paper with examples showing and summarizing some differences between complex matrices and quaternionic matrices and right and left eigenvalues of quaternionic matrices

An Update on a Few Permanent Conjectures

We review and update on a few conjectures concerning matrix permanent that are easily stated, understood, and accessible to general math audience. They are: Soules permanent-on-top conjecture†, Lieb permanent dominance conjecture, Bapat and Sunder conjecture† on Hadamard product and diagonal entries, Chollet conjecture on Hadamard product, Marcus conjecture on permanent of permanents, and several other conjectures. Some of these conjectures are recently settled; some are still open.We also raise a few new questions for future study. (†conjectures have been recently settled negatively.

Some inequalities for the eigenvalues of the product of positive semidefinite Hermitian matrices

AbstractLet λ1(A)⩾⋯⩾λn(A) denote the eigenvalues of a Hermitian n by n matrix A, and let 1⩽i1< ⋯ <ik⩽n. Our main results are ∑t=1kλt(GH)⩽∑t=1kλit(G)λn−it+1(H) and ∑t=1kλit(GH)⩽∑t=1kλit(G)λn−t+1(H). Here G and H are n by n positive semidefinite Hermitian matrices. These results extend Marshall and Olkin's inequality ∑t=1kλt(GH)⩽∑t=1kλt(G)λn−t+1(H). We also present analogous results for singular values

Harnack Inequalities: From Poincare Conjecture to Matrix Determinant

With a brief survey on the Harnack inequalities in various forms in Functional Analysis, in Partial Differential Equations, and in Perelman’s solution of the Poincare Conjecture, we discuss the Harnack inequality in Linear Algebra and Matrix Analysis. We present an extension of Tung’s inequality of Harnack type and study the equality case