Related papers: Spectrum maximizing products are not generically u…
Recently, Bochi and Laskawiec constructed an example of a set of matrices $\{A,B\}$ having two different (up to cyclic permutations of factors) spectrum maximizing products, $AABABB$ and $BBABAA$. In this paper, we identify a class of…
Experiments suggest that typical finite sets of square matrices admit spectrum maximizing products (SMPs): that is, products that attain the joint spectral radius (JSR). Furthermore, those SMPs are often combinatorially "simple." In this…
The joint spectral radius of a finite set of real $d \times d$ matrices is defined to be the maximum possible exponential rate of growth of long products of matrices drawn from that set. A set of matrices is said to have the…
We use an extension of the diagrammatic rules in random matrix theory to evaluate spectral properties of finite and infinite products of large complex matrices and large hermitian matrices. The infinite product case allows us to define a…
We address the problem of finite products that attain the joint spectral radius of a finite number of square matrices. Up to date the problem of existence of "forbidden products" remained open. We prove that the product $AABABABB$ (together…
The joint spectral radius of a bounded set of d times d real or complex matrices is defined to be the maximum exponential rate of growth of products of matrices drawn from that set. A set of matrices is said to satisfy the finiteness…
We consider products of $n$ random Hermitian matrices which generalize the one-matrix model and show its relation to Hurwitz numbers which count ramified coverings of certain type. Namely, these Hurwitz numbers count $2k$-fold ramified…
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…
Consider the product of $m$ independent $n\times n$ random matrices from the spherical ensemble for $m\ge 1$. The spectral radius is defined as the maximum absolute value of the $n$ eigenvalues of the product matrix. When $m=1$, the…
We consider m independent random rectangular matrices whose entries are independent and identically distributed standard complex Gaussian random variables. Assume the product of the m rectangular matrices is an n by n square matrix. The…
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…
The joint spectral radius of a bounded set of d times d real or complex matrices is defined to be the maximum exponential rate of growth of products of matrices drawn from that set. Under quite mild conditions such a set of matrices admits…
We study the spectrum of the product of two Toeplitz operators. Assume that the symbols of these operators are continuous and real-valued and that one of them is non-negative. We prove that the spectrum of the product of finite section…
This note deals with a simultaneous approximation of several matrices by a finite family of diagonalizable matrices satisfying an additional condition for the spectrum of a matrix product. That is the simplicity of all eigenvalues.
We investigate joint spectral characteristics of a family of matrices $\mathcal F $, associated with products in the semigroup generated by $\mathcal F$. In the literature, extremal measures such as the well-known joint spectral radius and…
Motivated by the conjectures formulated in 2003 by Tun\c{c}el et al., we study interlacing properties of the eigenvalues of $A\otimes B + B\otimes A$ for pairs of $n$-by-$n$ matrices $A, B$. We prove that for every pair of symmetric…
In this article we consider the maximum possible growth rate of sequences of long products of $d \times d$ matrices all of which are drawn from some specified compact set which has been normalised so as to have joint spectral radius equal…
In the paper, a simple condition guaranteing the finiteness property for a bounded set of matrices is presented. Given a bounded set S of real or complex matrices, it is shown that existence of a sequence of matrix products such that the…
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…
In 1969 H\"ochster proved that for every quasi-compact T1-space $X$ we can find a commutative ring $R$ such that $X$ is homeomorphic to the maximal spectrum $\mathrm{Specm}(R)$ of $R$. This result implies the existence of a commutative ring…