Related papers: An inequality for the matrix pressure function and…
We prove explicit polynomial bounds for Bochi's inequalities regarding the joint spectral radius of a subset of $d\times d$ matrices.
The lower spectral radius of a set of $d \times d$ matrices is defined to be the minimum possible exponential growth rate of long products of matrices drawn from that set. When considered as a function of a finite set of matrices of fixed…
In this paper we develop algorithms for approximating matrix multiplication with respect to the spectral norm. Let A\in{\RR^{n\times m}} and B\in\RR^{n \times p} be two matrices and \eps>0. We approximate the product A^\top B using two…
The affinity dimension is a number associated to an iterated function system of affine maps, which is fundamental in the study of the fractal dimensions of self-affine sets. De-Jun Feng and the author recently solved a folklore open…
We use ergodic theory to prove a quantitative version of a theorem of M. A. Berger and Y. Wang, which relates the joint spectral radius of a set of matrices to the spectral radii of finite products of those matrices. The proof rests on a…
In this paper we show a characterization of the joint spectral radius of a set of matrices as the limit of the $p$-radius of an associated probability distribution when $p$ tends to $\infty$. Allowing the set to have infinitely many…
Consider a square random matrix with independent and identically distributed entries of mean zero and unit variance. We show that as the dimension tends to infinity, the spectral radius is equivalent to the square root of the dimension in…
We prove that when the Aubry set for a Lipschitz continuous potential is a subshift of finite type, then the pressure function converges exponentially fast to its asymptote as the temperature goes to 0. The speed of convergence turns out to…
In this paper, we give upper and lower bounds for the spectral radius of a nonnegative irreducible matrix and characterize the equality cases. These bounds theoretically improve and generalize some known results of Duan et al.[X. Duan, B.…
The sub-additive pressure function $P(s)$ for an affine iterated function system (IFS) and the affinity dimension, defined as the unique solution $s_0$ to $P(s_0)=1$, were introduced by K. Falconer in his seminal 1988 paper on self-affine…
Let $(\Sigma_A, \sigma)$ be a subshift of finite type and let $M(x)$ be a continuous function on $\Sigma_A$ taking values in the set of non-negative matrices. We extend the classical scalar pressure function to this new setting and prove…
The pressure function is a fundamental object in various areas of mathematics. Its regularity is studied to derive insights into phase transitions in certain physical systems or to determine the Hausdorff dimension of self-affine sets. In…
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…
The joint spectral radius of a pair of 2x2 real matrices $(A_0,A_1)\in M_2(\mathbb{R})^2$ is defined to be $r(A_0,A_1)= \limsup_{n\to\infty} \max \{\|A_{i_1}...A_{i_n}\|^{1/n}: i_j\in\{0,1\}\}$, the optimal growth rate of the norm of…
The joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate that can be obtained by forming long products of matrices taken from the set. This quantity appears in a number of application contexts but is…
Low-rank pseudoinverses are widely used to approximate matrix inverses in scalable machine learning, optimization, and scientific computing. However, real-world matrices are often observed with noise, arising from sampling, sketching, and…
The famous Gelfand formula $\rho(A)= \limsup_{n\to\infty}\|A^{n}\|^{1/n}$ for the spectral radius of a matrix is of great importance in various mathematical constructions. Unfortunately, the range of applicability of this formula is…
Consider the ensembles of real symmetric Toeplitz matrices and real symmetric Hankel matrices whose entries are i.i.d. random variables chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments.…
The lower spectral radius, or joint spectral subradius, of a set of real $d \times d$ matrices is defined to be the smallest possible exponential growth rate of long products of matrices drawn from that set. The lower spectral radius arises…
In this paper, we study tail inequalities of the largest eigenvalue of a matrix infinitely divisible (i.d.) series, which is a finite sum of fixed matrices weighted by i.d. random variables. We obtain several types of tail inequalities,…