Related papers: A Fractal Eigenvector
Riordan matrices are infinite lower triangular matrices determined by a pair of formal power series over the real or complex field. These matrices have been mainly studied as combinatorial objects with an emphasis placed on the algebraic or…
We study the combinatorial and algebraic properties of Nonnegative Matrices. Our results are divided into three different categories. 1. We show a quantitative generalization of the 100 year-old Perron-Frobenius theorem, a fundamental…
For matrices with all nonnegative entries, the Perron-Frobenius theorem guarantees the existence of an eigenvector with all nonnegative components. We show that the existence of such an eigenvector is also guaranteed for a very different…
The Mandelbrot set is a fractal which classifies the behaviour of complex quadratic polynomials. Although its remarkably simple definition: $\mathcal{M}:=\{c \in \mathbb{C}\,|\,Q_c(0)^n \nrightarrow \infty \mbox{ as } n\rightarrow \infty,…
We consider the "Mandelbrot set" $M$ for pairs of complex linear maps, introduced by Barnsley and Harrington in 1985 and studied by Bousch, Bandt and others. It is defined as the set of parameters $\lambda$ in the unit disk such that the…
Consider $n$ linearly independent vectors in $\mathbb{C}^n$ which form columns of a matrix $A$. The recursive evaluation of eigen directions (normalized eigenvectors) of $A$ is the solution of an eigenvalue problem of the form…
Denote by $M_n(K)$ the algebra of $n$ by $n$ matrices with entries in the field $K$. A theorem of Albert and Muckenhoupt states that every trace zero matrix of $M_n(K)$ can be expressed as $AB-BA$ for some pair $(A,B)$ of matrices of…
In this note we describe the singular locus of diagonally-dominant Hermitian matrices with nonnegative diagonal entries over the reals, the complex numbers, and the quaternions. This yields explicit expressions for the probability that such…
A lot of formal and informal recreational study took place in the fields of Meromorphic Maps, since Mandelbrot popularized the map z <- z^2 + c. An immediate generalization of the Mandelbrot z <-z^n + c also known as the Multibrot family…
Let $k\leq n$ be positive integers and $\mathbb{Z}_{n}$ be the set of integers modulo $n$. A conjecture of Baranyai from 1974 asks for a decomposition of $k$-element subsets of $\mathbb{Z}_{n}$ into particular families of sets called…
Let $d$ and $n$ be integers satisfying $C\leq d\leq \exp(c\sqrt{\ln n})$ for some universal constants $c, C>0$, and let $z\in \mathbb{C}$. Denote by $M$ the adjacency matrix of a random $d$-regular directed graph on $n$ vertices. In this…
The famous MLC Conjecture states that the Mandelbrot set is locally connected, and it is considered by many to be the central conjecture in one-dimensional complex dynamics. Among others, it implies density of hyperbolicity in the quadratic…
We give a framework to study the connectedness of the set of zeros of power series with coefficients in a finite subset $G\subset \mathbb{C}$. We prove that the set of zeros in the unit disk is connected and locally connected if some graph…
Tubal scalars are usual vectors, and tubal matrices are matrices with every element being a tubal scalar. Such a matrix is often recognized as a third-order tensor. The product between tubal scalars, tubal vectors, and tubal matrices can be…
The set of mxn singular matrix pencils with normal rank at most r is an algebraic set with r+1 irreducible components. These components are the closure of the orbits (under strict equivalence) of r+1 matrix pencils which are in Kronecker…
We consider multiplicative semigroups of real dxd matrices. A semigroup S is called Perron if each of its matrices has a Perron eigenvalue, i.e., an eigenvalue equal to the spectral radius. If all matrices of S leave a proper convex cone…
The eigenvalue statistics of a pair $(M_1,M_2)$ of $n\times n$ Hermitian matrices taken random with respect to the measure $$\frac{1}{Z_n}\exp\big(-n\Tr (V(M_1)+W(M_2)-\tau M_1M_2)\big) {\rm d}M_1 {\rm d} M_2 $$ can be described in terms of…
Statistical properties of non--symmetric real random matrices of size $M$, obtained as truncations of random orthogonal $N\times N$ matrices are investigated. We derive an exact formula for the density of eigenvalues which consists of two…
In the recent paper \cite{1}, Denton et al. provided the eigenvector-eigenvalue identity for Hermitian matrices, and a survey was also given for such identity in the literature. The main aim of this paper is to present the identity related…
If $A$ is an $n \times n$ Hermitian matrix with eigenvalues $\lambda_1(A),\dots,\lambda_n(A)$ and $i,j = 1,\dots,n$, then the $j^{\mathrm{th}}$ component $v_{i,j}$ of a unit eigenvector $v_i$ associated to the eigenvalue $\lambda_i(A)$ is…