Related papers: Norm inequalities involving geometric means

We consider the set Bp of parametric block correlation matrices with p blocks of various (and possibly different) sizes, whose diagonal blocks are compound symmetry (CS) correlation matrices and off-diagonal blocks are constant matrices.…

We prove that for any semi-norm $\|\cdot\|$ on $\mathbb{R}^n,$ and any symmetric convex body $K$ in $\mathbb{R}^n,$ \begin{equation}\label{ineq-abs2} \int_{\partial K} \frac{\|n_x\|^2}{\langle x,n_x\rangle}\leq…

In this note we prove results of the following types. Let be given distinct complex numbers $z_j$ satisfying the conditions $|z_j| = 1, z_j \not= 1$ for $j=1,..., n$ and for every $z_j$ there exists an $ i$ such that $z_i = \bar{z_j}. $…

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…

Let $d\geq 2$ and $k\geq 1$ be fixed. We prove that, for every $\epsilon>0$ and every real $\beta$, there exist integers $1\leq b_1,\ldots,b_k\leq N$ such that \[ \left\|\sum_{j=1}^k b_j^{1/d}-\beta\right\| \ll_{d,k,\epsilon}…

We prove that every infinite sequence of skew-symmetric or symmetric matrices M_1, M_2, ... over a fixed finite field must have a pair M_i, M_j (i<j) such that M_i is isomorphic to a principal submatrix of the Schur complement of a…

Let $A_1, \ldots ,A_m$ and $B_1, \ldots ,B_m$ be subsets of $[n]$ and let $t$ be a non-negative integer with the following property: $|A_i \cap B_i|\leq t$ for each $i$ and $|A_i\cap B_j|>t$ whenever $i< j$. Then $m\leq 2^{n-t}$. Our proof…

We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as an infinite sum of geometric terms. We present necessary conditions for the invariant measure…

Let $P$ be an $m$-homogeneous polynomial in $n$-complex variables $x_1, \dotsc, x_n$. Clearly, $P$ has a unique representation in the form \begin{equation*} P(x)= \sum_{1 \leq j_1 \leq \dotsc \leq j_m \leq n} c_{(j_1, \dotsc, j_m)} \,…

The Hardy--Littlewood inequalities for multilinear forms on sequence spaces state that for all positive integers $m,n\geq2$ and all $m$-linear forms $T:\ell_{p_{1}}^{n}\times\cdots\times\ell_{p_{m}}^{n}\rightarrow\mathbb{K}$…

We study matrix inequalities involving partial traces for positive semidefinite block matrices. First of all, we present a new method to prove a celebrated result of Choi [Linear Algebra Appl. 516 (2017)]. Our method also allows us to prove…

For positive semidefinite $n\times n$ matrices $A$ and $B$, the singular value inequality $(2+t)s_{j}(A^{r}B^{2-r}+A^{2-r}B^{r})\leq 2s_{j}(A^{2}+tAB+B^{2})$ is shown to hold for $r=\frac{1}{2}, 1, \frac{3}{2}$ and all $-2<t\leq 2$.

We prove Hunter's conjecture on complete homogeneous symmetric polynomials. For even $n$ and every integer $k\geq 1$, we show that under the constraint $\sum_{i=1}^n a_i^2=1$ the global minimum of the even-degree polynomial…

The exact/approximate non-orthogonal general joint block diagonalization ({\sc nogjbd}) problem of a given real matrix set $\mathcal{A}=\{A_i\}_{i=1}^m$ is to find a nonsingular matrix $W\in\mathbb{R}^{n\times n}$ (diagonalizer) such that…

A conjecture posed by S. Hayajneh and F. Kittaneh claims that given $A,B$ positive matrices, $0\le t\le 1$, and any unitarily invariant norm it holds $|||A^tB^{1-t}+B^tA^{1-t}|||\le|||A^tB^{1-t}+A^{1-t}B^t|||$. Recently, R. Bhatia proved…

Suppose that $C$ is a bounded, convex subset of $\mathbb{R}^n$, and that $P_1, \dots, P_k$ are planks which cover $C$ in respective directions $v_1, \dots, v_k$ and with widths $w_1, \dots, w_k$. In 1951, Bang conjectured that the sum of…

Let $\|\cdot\|_{\mathbf A}$ be a norm on $\mathbb C^m$ given by the formula $\|(z_1,\ldots,z_m)\|_{\mathbf A}=\|z_1A_1+\cdots+z_mA_m\|_{\rm op}$ for some choice of an $m$-tuple of $n\times n$ linearly independent matrices $\mathbf A=(A_1,…

Given two symmetric and positive semidefinite square matrices $A, B$, is it true that any matrix given as the product of $m$ copies of $A$ and $n$ copies of $B$ in a particular sequence must be dominated in the spectral norm by the ordered…

The Steinitz lemma, a classic from 1913, states that $a_1,\ldots,a_n$, a sequence of vectors in $\R^d$ with $\sum_1^n a_i=0$, can be rearranged so that every partial sum of the rearranged sequence has norm at most $2d\max \|a_i\|$. In the…

We investigate the representation of symmetric polynomials as a sum of squares. Since this task is solved using semidefinite programming tools we explore the geometric, algebraic, and computational implications of the presence of discrete…