Lecture : Basic Matrix Results ( 2 of 3 )
نویسنده
چکیده
Today, we will consider a different class of matrices and establish a different type of “niceness” result, which will also be used extensively by spectral methods. In particular, we want to say something about how eigenvectors, and in particular the extremal eigenvectors, e.g., the largest one or few or the smallest one of few “look like.” The reason is that spectral methods—both vanilla and non-vanilla variants—will rely crucially on this; thus, understanding when and why this is true will be helpful to see how spectral methods sit with respect to other types of methods, to understand when they can be generalized, or not, and so on.
منابع مشابه
Statistical Mechanics and Random Matrices
Statistical Mechanics and Random Matrices 3 1. Introduction 6 2. Motivations 7 3. The different scales; typical results 12 Lecture 1. Wigner matrices and moments estimates 15 1. Wigner's theorem 16 2. Words in several independent Wigner matrices 23 3. Estimates on the largest eigenvalue of Wigner matrices 25 Lecture 2. Gaussian Wigner matrices and Fredholm determinants 27 1. Joint law of the ei...
متن کاملTeaching Nursing Students about the Basic Principles of Infection Control: Programmed Instruction or Lecture Method
Introduction. Programmed instruction (PI) and lecture based learning have been compared in previous studies in the world, but due to special condition of education in nursing, the effects of these two methods have been less investigated in Iranian universities. This study attempts to compare the effect of these two educational methods on nursing students' learning about the basic principles of ...
متن کاملCombinatorics of Triangulations and Hilbert Series
We introduce the basic concepts of Gröbner basis theory and its relations to polytope theory. This will cover very basic parts of [1], the basic chapters from [3] and the first chapters from [11]. In particular, we will define the Gröbner fan, which will play a major role in some of the research problems we will pose later. In this lecture we relate triangulations to Gröbner bases and Hilbert-s...
متن کاملLecture 4: Covering the Large Spectrum via Dual-sparse Approximation
1 Discrete Fourier analysis In this lecture, we use the dual-sparse approximation theorem from the last lecture to prove some results in discrete Fourier analysis. For simplicity, we restrict ourselves to the setting of G ¿ n 2 , but the theorems hold (when suitably restated) for any finite abelian group G. Fourier analysis over ¿ n 2. We use ¿ 2 {0, 1} to denote the field on two elements. Let ...
متن کاملAn Introduction to Additive Combinatorics
This is a slightly expanded write-up of my three lectures at the Additive Combinatorics school. In the first lecture we introduce some of the basic material in Additive Combinatorics, and in the next two lectures we prove two of the key background results, the Freiman-Ruzsa theorem and Roth’s theorem for 3-term arithmetic progressions. Lecture I: Introductory material 1. Basic Definitions. 2. I...
متن کاملLes Houches lectures on matrix models and topological strings
In these lecture notes for the Les Houches School on Applications of Random Matrices in Physics we give an introduction to the connections between matrix models and topological strings. We first review some basic results of matrix model technology and then we focus on type B topological strings. We present the main results of Dijkgraaf and Vafa describing the spacetime string dynamics on certai...
متن کامل