Related papers: On the minimum trace norm of (0,1)-matrices

We obtain tight lower bounds for the trace norm $\Vert \cdot \Vert_1$ of some matrices with diagonal zero, in terms of the entry-wise $L^1$-norm (denoted by $\Vert \cdot \Vert_{(1)}$). It is shown that on the space of nonzero real symmetric…

We study the spectral norm of matrices M that can be factored as M=BA, where A is a random matrix with independent mean zero entries, and B is a fixed matrix. Under the (4+epsilon)-th moment assumption on the entries of A, we show that the…

Let $\mathcal{S}$ be the set of all positive-definite, symmetrizable integer matrices with non-zero upper and lower diagonal and $\mathcal{T}$ to be the set of all positive-definite real symmetric matrices with nonzero upper diagonal such…

Let $D$ be a digraph of order $n$ with adjacency matrix $A(D)$. For $\alpha\in[0,1)$, the $A_{\alpha}$ matrix of $D$ is defined as $A_{\alpha}(D)=\alpha {\Delta}^{+}(D)+(1-\alpha)A(D)$, where…

Let $M$ be an $n\times n$ random matrix with entries in $\{0, 1\}$, where each row is independently and uniformly sampled from the set of all vectors in $\{0, 1\}^n$ containing exactly $d$ ones, with $d=pn$ for some fixed constant $p\in…

We generalise the formula expressing the matrix trace of a given square matrix as the integral of the numerical values of $A$ over the Euclidean sphere to the unit spheres of finite-dimensional normed spaces that have a 1-symmetric basis.…

Let $A$ be an $n\times n$ real matrix, and let $M$ be an $n\times n$ random matrix whose entries are i.i.d sub-Gaussian random variables with mean $0$ and variance $1$. We make two contributions to the study of $s_n(A+M)$, the smallest…

We show that the distance in total variation between $(\mathrm{Tr}\ U, \frac{1}{\sqrt{2}}\mathrm{Tr}\ U^2, \cdots, \frac{1}{\sqrt{m}}\mathrm{Tr}\ U^m)$ and a real Gaussian vector, where $U$ is a Haar distributed orthogonal or symplectic…

We prove the following proposition. Under linear interpolation on a Euclidean $n$-dimensional ball $B$, an interpolation projector whose nodes coincide with the vertices of a regular simplex inscribed into the boundary sphere has the…

In this note we prove that Tr (MN+ PQ)>= 0 when the following two conditions are met: (i) the matrices M, N, P, Q are structured as follows: M = A -B, N = inv(B)-inv(A), P = C-D, Q =inv (B+D)-inv(A+C), where inv(X) denotes the inverse…

Let $A \in \mathbb{R}^{N \times n}$ ($N \geq n$) be a random matrix with with independent entries that have mean 0 variance 1 and bounded $2+\beta$ moment. We show that the smallest singular value $\sigma_n(A)$ satisfies \[ \Pr…

Let $A$ be a $n \times n$ symmetric matrix with $(A_{i,j})_{i\leq j} $, independent and identically distributed according to a subgaussian distribution. We show that $$\mathbb{P}(\sigma_{\min}(A) \leq \varepsilon/\sqrt{n}) \leq C…

The Crouzeix ratio $\psi(A)$ of an $N\times N$ complex matrix $A$ is the supremum of $\|p(A)\|$ taken over all polynomials $p$ such that $|p|\le 1$ on the numerical range of $A$. It is known that $\psi(A)\le 1+\sqrt{2}$, and it is…

Let $M$ be an $n\times n$ random i.i.d. matrix. This paper studies the deviation inequality of $s_{n-k+1}(M)$, the $k$-th smallest singular value of $M$. In particular, when the entries of $M$ are subgaussian, we show that for any…

Let $\Phi(N)$ denote the number of products of matrices $[ \begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}]$ and $[ \begin{smallmatrix} 1 & 0 \\ 1 & 1 \end{smallmatrix} ]$ of trace equal to $N$, and $\Psi(N)=\sum_{n=3}^N \Phi(n)$ be…

A sign pattern matrix is a matrix whose entries are from the set $\{+,-,0\}$. If $A$ is an $m\times n$ sign pattern matrix, the qualitative class of $A$, denoted $Q(A)$, is the set of all real $m\times n$ matrices $B=[b_{i,j}]$ with…

Let $\log^{2+\varepsilon} n \le d \le n/2$ for some fixed $\varepsilon \in (0,1)$, and let $M_n$ be an $n\times n$ random matrix with entries in ${0,1}$, where each row is independently and uniformly sampled from the set of all vectors in…

The trace norm $\left\Vert G\right\Vert _{\ast}$ of a graph $G$ is the sum of its singular values, i.e., the absolute values of its eigenvalues. The norm $\left\Vert G\right\Vert _{\ast}$ has been intensively studied under the name of graph…

In this article, we obtain a super-exponential rate of convergence in total variation between the traces of the first $m$ powers of an $n\times n$ random unitary matrices and a $2m$-dimensional Gaussian random variable. This generalizes…

We present a formula for the trace of any symmetric power of a $n\times n$ matrix (with coefficients in a field) in terms of the ordinary powers of the matrix, an arbitrarily chosen linear function which vanishes on the identity matrix, and…