Related papers: On matrices for which norm bounds are attained

The positive semidefinite rank of a nonnegative $(m\times n)$-matrix~$S$ is the minimum number~$q$ such that there exist positive semidefinite $(q\times q)$-matrices $A_1,\dots,A_m$, $B_1,\dots,B_n$ such that $S(k,\ell) = \mbox{tr}(A_k^*…

We considered the problem of obtaining upper bounds for the mathematical expectation of the $q$-norm ($2\leqslant q \leqslant \infty$) of the vector which is uniformly distributed on the unit Euclidean sphere. We finish the paper with…

We provide numerical bounds on the Crouzeix ratiofor KLS matrices $A$ which have a line segment on the boundary of the numerical range. The Crouzeix ratio is the supremum over all polynomials $p$ of the spectral norm of $p(A)$ dividedby the…

Let $A_q(n,d)$ be the maximum order (maximum number of codewords) of a $q$-ary code of length $n$ and Hamming distance at least $d$. And let $A(n,d,w)$ that of a binary code of constant weight $w$. Building on results from algebraic graph…

In numerical analysis it is often necessary to estimate the condition number $CN(T)=||T||_{} \cdot||T^{-1}||_{}$ and the norm of the resolvent $||(\zeta-T)^{-1}||_{}$ of a given $n\times n$ matrix $T$. We derive new spectral estimates for…

This paper proposes lower bounds on a quantity called $L^p$-norm joint spectral radius, or in short, $p$-radius, of a finite set of matrices. Despite its wide range of applications to, for example, stability analysis of switched linear…

We construct simple, explicit matrices with columns having unit $\ell^2$ norm and discrepancy approaching $1 + \sqrt{2} \approx 2.414$. This number gives a lower bound, the strongest known as far as we are aware, on the constant appearing…

Motivated by the least squares solver Blendenpik, we investigate three strategies for uniform sampling of rows from m x n matrices Q with orthonormal columns. The goal is to determine, with high probability, how many rows are required so…

The aim of this paper is to present new upper bounds for the distance between a properly normalized permanent of a rectangular complex matrix and the product of the arithmetic means of the entries of its columns. It turns out that the…

The 0-1 matrix A contains a 0-1 matrix M if some submatrix of A can be transformed into M by changing some ones to zeroes. If A does not contain M, then A avoids M. Let ex(n,M) be the maximum number of ones in an n x n 0-1 matrix that…

Given integers $p\ge q>1$, a family of sets satisfies the $(p,q)$ property if among any $p$ members of it some $q$ intersect. We prove that for any fixed integer constants $p\ge q>1$, a family of $d$-intervals satisfying the $(p,q)$…

We find an upper bound for the number of groups of order $n$ up to isomorphism in the variety $G = A_pA_qA_r$, where $p$, $q$ and $r$ are distinct primes. We also find a bound on the orders and on the number of conjugacy classes of…

We prove that $\det A\leq 6^\frac{n}{6}$ whenever $A\in\{0,1\}^{n\times n}$ contains at most $2n$ ones. We also prove an upper bound on the determinant of matrices with the $k$-consecutive ones property, a generalisation of the consecutive…

Let $F$ be a $k\times \ell$ (0,1)-matrix. Define a (0,1)-matrix $A$ to have a $F$ as a \emph{configuration} if there is a submatrix of $A$ which is a row and column permutation of $F$. In the language of sets, a configuration is a…

We derive upper and lower bounds on the determinant of an exponential matrix. They can be transformed into corresponding bounds for the determinant of a univariate Gaussian matrix.

In a celebrated paper of 1893, Hadamard established the maximal determinant theorem, which establishes an upper bound on the determinant of a matrix with complex entries of norm at most $1$. His paper concludes with the suggestion that…

A central question in random matrix theory is universality. When an emergent phenomena is observed from a large collection of chosen random variables it is natural to ask if this behavior is specific to the chosen random variable or if the…

We present a preliminary study of Schur norms $\|M\|_{S}=\max\{ \|M\circ C\|: \|C\|=1\}$, where M is a matrix whose entries are $\pm1$, and $\circ$ denotes the entrywise (i.e., Schur or Hadamard) product of the matrices. We show that, if…

We prove that the known sufficient conditions on the real parameters $(p,q)$ for which the matrix power mean inequality $((A^p+B^p)/2)^{1/p}\le((A^q+B^q)/2)^{1/q}$ holds for every pair of matrices $A,B>0$ are indeed best possible. The proof…

We prove the existence of a positive semidefinite matrix $A \in \mathbb{R}^{n \times n}$ such that any decomposition into rank-1 matrices has to have factors with a large $\ell^1-$norm, more precisely $$ \sum_{k} x_k x_k^*=A \quad \implies…