Trace and Eigenvalue Inequalities of Ordinary and Hadamard Products for Positive Semidefinite Hermitian Matrices

Let Aand Bbe $n \times n$ positive semidefinite Hermitian matrices, let $\alpha$ and $\beta$ be real numbers, let $\circ$ denote the Hadamard product of matrices, and let $A_k$ denote any $k \times k$ principal submatrix of A. The following trace and eigenvalue inequalities are shown: $$\operatorname{tr}(A \circ B)^\alpha \leq \operatorname{tr}(A^\alpha \circ B^\alpha ),\quad\alpha \leq 0\,{\text{ or }}\,\alpha \geq 1,$$$$\operatorname{tr}(A \circ B)^\alpha \geq \operatorname{tr}(A^\alpha \circ B^\alpha ),\quad 0 \leq \alpha \leq 1,$$$$\lambda^{1/ \alpha } (A^\alpha \circ B^\alpha ) \leq \lambda ^{1/\beta } (A^\beta \circ B^\beta ),\quad\alpha \leq \beta ,\alpha \beta \ne 0,$$$$\lambda ^{1/\alpha } [(A^\alpha )_k ] \leq \lambda ^{1/\beta } [(A^\beta )_k ],\quad\alpha \leq \beta ,\alpha \beta \ne 0.$$The equalities corresponding to the inequalities above and the known inequalities $$\operatorname{tr}(AB)^\alpha \leq \operatorname{tr}(A^\alpha B^\alpha ),\quad | \alpha | \geq 1,$$ and $$\operatorname{tr}(AB)^\alpha \geq \operatorname{tr}(A^\alpha B^\alpha ),\quad | \alpha | \leq 1$$ are thoroughly discussed. Some applications are given

Some Bounds For The Spectral Radius Of Hadamard Product & Kronecker Product Of Matrices.

The main aim of this study was to discuss some bounds for the Spectral Radius of the Hadamard Product of matrices. This study presents several spectral radius inequalities for sums, product ( hadamard product), and comutators of matrices, and it exposes to some properties of the hadamard product and the relationship between hadamard product and kronecker product for spectral radius of matrix. Applications of these results are also given. Keywords: Spectral Radius, Hadamard product, Kronecker Produc

The Expected Norm of a Sum of Independent Random Matrices: An Elementary Approach

In contemporary applied and computational mathematics, a frequent challenge is to bound the expectation of the spectral norm of a sum of independent random matrices. This quantity is controlled by the norm of the expected square of the random matrix and the expectation of the maximum squared norm achieved by one of the summands; there is also a weak dependence on the dimension of the random matrix. The purpose of this paper is to give a complete, elementary proof of this important, but underappreciated, inequality.Comment: 20 page

Trace inequalities in nonextensive statistical mechanics

In this short paper, we establish a variational expression of the Tsallis relative entropy. In addition, we derive a generalized thermodynamic inequality and a generalized Peierls-Bogoliubov inequality. Finally we give a generalized Golden-Thompson inequality

Second-Order Matrix Concentration Inequalities

Matrix concentration inequalities give bounds for the spectral-norm deviation of a random matrix from its expected value. These results have a weak dimensional dependence that is sometimes, but not always, necessary. This paper identifies one of the sources of the dimensional term and exploits this insight to develop sharper matrix concentration inequalities. In particular, this analysis delivers two refinements of the matrix Khintchine inequality that use information beyond the matrix variance to reduce or eliminate the dimensional dependence.Comment: 27 pages. Revision corrects technical errors in several place